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Chapter 7: Confidence Intervals

Chapter 7 Homework

7.1 homework.

Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish to construct a 95% confidence interval for the mean height of male Swedes. Forty-eight male Swedes are surveyed. The sample mean is 71 inches. The sample standard deviation is 2.8 inches.

Construct a 95% confidence interval for the population mean height of male Swedes.

homework section 7 2 stats

Announcements for 84 upcoming engineering conferences were randomly picked from a stack of IEEE Spectrum magazines. The mean length of the conferences was 3.94 days, with a standard deviation of 1.28 days. Assume the underlying population is normal.

  • In words, define the random variables X and [latex]\overline{X}[/latex].
  • Which distribution should you use for this problem? Explain your choice.
  • State the confidence interval.
  • Sketch the graph.
  • Calculate the error bound.

Suppose that an accounting firm does a study to determine the time needed to complete one person’s tax forms. It randomly surveys 100 people. The sample mean is 23.6 hours. There is a known standard deviation of 7.0 hours. The population distribution is assumed to be normal.

  • [latex]\overline{x}[/latex] =________
  • σ =________
  • n =________
  • If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?
  • If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Why?
  • Suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within one hour. How would the number of people the firm surveys change? Why?
  • [latex]\overline{x}[/latex] = 23.6
  • [latex]\sigma[/latex] = 7
  • X is the time needed to complete an individual tax form. [latex]\overline{X}[/latex] is the mean time to complete tax forms from a sample of 100 customers.
  • (22.228, 24.972)

homework section 7 2 stats

  • EBM = 1.372
  • It will need to change the sample size. The firm needs to determine what the confidence level should be, then apply the error bound formula to determine the necessary sample size.
  • The confidence level would increase as a result of a larger interval. Smaller sample sizes result in more variability. To capture the true population mean, we need to have a larger interval.
  • According to the error bound formula, the firm needs to survey 206 people. Since we increase the confidence level, we need to increase either our error bound or the sample size.

A sample of 16 small bags of the same brand of candies was selected. Assume that the population distribution of bag weights is normal. The weight of each bag was then recorded. The mean weight was two ounces with a standard deviation of 0.12 ounces. The population standard deviation is known to be 0.1 ounce.

  • s x =________
  • In words, define the random variable X .
  • In words, define the random variable [latex]\overline{X}[/latex].
  • In complete sentences, explain why the confidence interval in part f is larger than the confidence interval in part e.
  • In complete sentences, give an interpretation of what the interval in part f means.

A camp director is interested in the mean number of letters each child sends during his or her camp session. The population standard deviation is known to be 2.5. A survey of 20 campers is taken. The mean from the sample is 7.9 with a sample standard deviation of 2.8.

  • Define the random variables X and [latex]\overline{X}[/latex] in words.
  • What will happen to the error bound and confidence interval if 500 campers are surveyed? Why?

N [latex]7.9\left(\frac{2.5}{\sqrt{20}}\right)[/latex]

homework section 7 2 stats

What is meant by the term “90% confident” when constructing a confidence interval for a mean?

  • If we took repeated samples, approximately 90% of the samples would produce the same confidence interval.
  • If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean.
  • If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean.
  • If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples.

The Federal Election Commission collects information about campaign contributions and disbursements for candidates and political committees each election cycle. During the 2012 campaign season, there were 1,619 candidates for the House of Representatives across the United States who received contributions from individuals. [link] shows the total receipts from individuals for a random selection of 40 House candidates rounded to the nearest 💲100. The standard deviation for this data to the nearest hundred is σ = 💲909,200.

  • Find the point estimate for the population mean.
  • Using 95% confidence, calculate the error bound.
  • Create a 95% confidence interval for the mean total individual contributions.
  • Interpret the confidence interval in the context of the problem.
  • [latex]\overline{x}[/latex] = 💲568,873

EBM = [latex]{z}_{0.025}\frac{\sigma }{\sqrt{n}}[/latex] = 1.96 [latex]\frac{909200}{\sqrt{40}}[/latex] = 💲281,764

[latex]\overline{x}[/latex] + EBM = 568,873 + 281,764 = 850,637

Alternate solution:

Press STAT and arrow over to TESTS .

  • Arrow down to 7:ZInterval .

Press ENTER .

  • Arrow to Stats and press ENTER .
  • σ : 909,200
  • [latex]\overline{x}[/latex]: 568,873
  • Arrow down to Calculate and press ENTER .
  • The confidence interval is (💲287,114, 💲850,632).
  • Notice the small difference between the two solutions—these differences are simply due to rounding errors in the hand calculations.
  • We estimate with 95% confidence that the mean amount of contributions received from all individuals by House candidates is between 💲287,109 and 💲850,637.

The American Community Survey (ACS), part of the United States Census Bureau, conducts a yearly census similar to the one taken every ten years, but with a smaller percentage of participants. The most recent survey estimates with 90% confidence that the mean household income in the U.S. falls between 💲69,720 and 💲69,922. Find the point estimate for mean U.S. household income and the error bound for mean U.S. household income.

The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. You want to estimate the mean height of students at your college or university to within one inch with 93% confidence. How many male students must you measure?

Use the formula for EBM , solved for n :

[latex]n= \frac{{z}^{2}{\sigma }^{2}}{EB{M}^{2}}[/latex]

From the statement of the problem, you know that σ = 2.5, and you need EBM = 1.

z = z 0.035 = 1.812

(This is the value of z for which the area under the density curve to the right of z is 0.035.)

[latex]n= \frac{{z}^{2}{\sigma }^{2}}{EB{M}^{2}}=\frac{{1.812}^{2}{2.5}^{2}}{{1}^{2}} \approx  20.52[/latex]

You need to measure at least 21 male students to achieve your goal.

7.2 Homework

In six packages of “The Flintstones® Real Fruit Snacks” there were five Bam-Bam snack pieces. The total number of snack pieces in the six bags was 68. We wish to calculate a 96% confidence interval for the population proportion of Bam-Bam snack pieces.

  • Define the random variables X and P ′ in words.
  • Which distribution should you use for this problem? Explain your choice
  • Calculate p ′.
  • Do you think that six packages of fruit snacks yield enough data to give accurate results? Why or why not?

A random survey of enrollment at 35 community colleges across the United States yielded the following figures: 6,414; 1,550; 2,109; 9,350; 21,828; 4,300; 5,944; 5,722; 2,825; 2,044; 5,481; 5,200; 5,853; 2,750; 10,012; 6,357; 27,000; 9,414; 7,681; 3,200; 17,500; 9,200; 7,380; 18,314; 6,557; 13,713; 17,768; 7,493; 2,771; 2,861; 1,263; 7,285; 28,165; 5,080; 11,622. Assume the underlying population is normal.

Construct a 95% confidence interval for the population mean enrollment at community colleges in the United States.

  • CI: (6244, 11,014)

homework section 7 2 stats

  • It will become smaller

Suppose that a committee is studying whether or not there is waste of time in our judicial system. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. The committee randomly surveyed 81 people who recently served as jurors. The sample mean wait time was eight hours with a sample standard deviation of four hours.

  • [latex]\overline{x}[/latex] = __________
  • [latex]{s}_{x}[/latex] = __________
  • n = __________
  • n – 1 = __________
  • Define the random variables [latex]X[/latex] and [latex]\overline{X}[/latex] in words.
  • Explain in a complete sentence what the confidence interval means.

A pharmaceutical company makes tranquilizers. It is assumed that the distribution for the length of time they last is approximately normal. Researchers in a hospital used the drug on a random sample of nine patients. The effective period of the tranquilizer for each patient (in hours) was as follows: 2.7; 2.8; 3.0; 2.3; 2.3; 2.2; 2.8; 2.1; and 2.4.

  • Define the random variable [latex]X[/latex] in words.
  • Define the random variable [latex]\overline{X}[/latex] in words.
  • What does it mean to be “95% confident” in this problem?
  • [latex]\overline{x}[/latex] = 2.51
  • [latex]{s}_{x}[/latex] = 0.318
  • the effective length of time for a tranquilizer
  • the mean effective length of time of tranquilizers from a sample of nine patients
  • CI: (2.27, 2.76)
  • Check student’s solution.
  • If we were to sample many groups of nine patients, 95% of the samples would contain the true population mean length of time.

Suppose that 14 children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. It was revealed that they used them an average of six months with a sample standard deviation of three months. Assume that the underlying population distribution is normal.

  • Define the random variable[latex]\overline{X}[/latex] in words.
  • Why would the error bound change if the confidence level were lowered to 90%?

The Federal Election Commission (FEC) collects information about campaign contributions and disbursements for candidates and political committees each election cycle. A political action committee (PAC) is a committee formed to raise money for candidates and campaigns. A Leadership PAC is a PAC formed by a federal politician (senator or representative) to raise money to help other candidates’ campaigns.

The FEC has reported financial information for 556 Leadership PACs that operated during the 2011–2012 election cycle. The following table shows the total receipts during this cycle for a random selection of 20 Leadership PACs.

[latex]\overline{x}=💲251,854.23[/latex]

[latex]s=\text{ }💲521,130.41[/latex]

Use this sample data to construct a 96% confidence interval for the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle. Use the Student’s t-distribution.

Note that we are not given the population standard deviation, only the standard deviation of the sample.

There are 30 measures in the sample, so n = 30, and df = 30 – 1 = 29

CL = 0.96, so α = 1 – CL = 1 – 0.96 = 0.04

[latex]\frac{\alpha }{2}=0.02{t}_{\frac{\alpha }{2}}={t}_{0.02}[/latex] = 2.150

[latex]EBM={t}_{\frac{\alpha }{2}}\left(\frac{s}{\sqrt{n}}\right)=2.150\left(\frac{521,130.41}{\sqrt{30}}\right) ~ 💲204,561.66[/latex]

[latex]\overline{x}[/latex] – EBM = 💲251,854.23 – 💲204,561.66 = 💲47,292.57

[latex]\overline{x}[/latex] + EBM = 💲251,854.23+ 💲204,561.66 = 💲456,415.89

We estimate with 96% confidence that the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle lies between 💲47,292.57 and 💲456,415.89.

Alternate Solution

Enter the data as a list.

Arrow down to 8:TInterval .

Arrow to Data and press ENTER .

Arrow down and enter the name of the list where the data is stored.

Enter Freq : 1

Enter C-Level : 0.96

Arrow down to Calculate and press Enter .

The 96% confidence interval is (💲47,262, 💲456,447).

The difference between solutions arises from rounding differences.

Forbes magazine published data on the best small firms in 2012. These were firms that had been publicly traded for at least a year, have a stock price of at least 💲5 per share, and have reported annual revenue between 💲5 million and 💲1 billion. The [link] shows the ages of the corporate CEOs for a random sample of these firms.

Use this sample data to construct a 90% confidence interval for the mean age of CEO’s for these top small firms. Use the Student's t-distribution.

Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its mean number of unoccupied seats per flight over the past year. To accomplish this, the records of 225 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats.

  • n -1 = __________
  • Construct a 92% confidence interval for the population mean number of unoccupied seats per flight.
  • [latex]\overline{x}[/latex] =
  • [latex]{s}_{x}[/latex] =
  • X is the number of unoccupied seats on a single flight. [latex]\overline{X}[/latex] is the mean number of unoccupied seats from a sample of 225 flights.
  • CI: (11.12 , 12.08)

In a recent sample of 84 used car sales costs, the sample mean was 💲6,425 with a standard deviation of 💲3,156. Assume the underlying distribution is approximately normal.

  • Explain what a “95% confidence interval” means for this study.

Six different national brands of chocolate chip cookies were randomly selected at the supermarket. The grams of fat per serving are as follows: 8; 8; 10; 7; 9; 9. Assume the underlying distribution is approximately normal.

  • If you wanted a smaller error bound while keeping the same level of confidence, what should have been changed in the study before it was done?
  • Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies.
  • Calculate the mean.
  • Is the mean within the interval you calculated in part a? Did you expect it to be? Why or why not?
  • CI: (7.64 , 9.36)

homework section 7 2 stats

  • The sample should have been increased.
  • Answers will vary.

A survey of the mean number of cents off that coupons give was conducted by randomly surveying one coupon per page from the coupon sections of a recent San Jose Mercury News. The following data were collected: 20¢; 75¢; 50¢; 65¢; 30¢; 55¢; 40¢; 40¢; 30¢; 55¢; 💲1.50; 40¢; 65¢; 40¢. Assume the underlying distribution is approximately normal.

  • If many random samples were taken of size 14, what percent of the confidence intervals constructed should contain the population mean worth of coupons? Explain why.

Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. serving size. The sample mean is 13.30 with a sample standard deviation of 1.55. Assume the underlying population is normally distributed.

Find the 95% Confidence Interval for the true population mean for the amount of soda served.

  • (12.42, 14.18)
  • (12.32, 14.29)
  • (12.50, 14.10)
  • Impossible to determine

What is the error bound?

7.3 Homework

Insurance companies are interested in knowing the population percentage of drivers who always buckle up before riding in a car.

  • When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within 0.03?
  • If it were later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?
  • The sample size would need to be increased since the critical value increases as the confidence level increases.

Suppose that the insurance companies did do a survey. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. We are interested in the population proportion of drivers who claim they always buckle up.

  • x = __________
  • p ′ = __________
  • Define the random variables X and P ′, in words.
  • If this survey were done by telephone, list three difficulties the companies might have in obtaining random results.

According to a recent survey of 1,200 people, 61% feel that the president is doing an acceptable job. We are interested in the population proportion of people who feel the president is doing an acceptable job.

X = the number of people who feel that the president is doing an acceptable job;

P ′ = the proportion of people in a sample who feel that the president is doing an acceptable job.

  • CI: (0.59, 0.63)
  • Check student’s solution

An article regarding interracial dating and marriage recently appeared in the Washington Post . Of the 1,709 randomly selected adults, 315 identified themselves as Latinos, 323 identified themselves as blacks, 254 identified themselves as Asians, and 779 identified themselves as whites. In this survey, 86% of blacks said that they would welcome a white person into their families. Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person.

  • We are interested in finding the 95% confidence interval for the percent of all black adults who would welcome a white person into their families. Define the random variables X and P ′, in words.

Refer to the information in [link] .

  • percent of all Asians who would welcome a white person into their families.
  • percent of all Asians who would welcome a Latino into their families.
  • percent of all Asians who would welcome a black person into their families.
  • Even though the three-point estimates are different, do any of the confidence intervals overlap? Which?
  • For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions?
  • For any intervals that do not overlap, in words, what does this imply about the significance of the differences in the true proportions?
  • (0.72, 0.82)
  • (0.65, 0.76)
  • (0.60, 0.72)
  • Yes, the intervals (0.72, 0.82) and (0.65, 0.76) overlap, and the intervals (0.65, 0.76) and (0.60, 0.72) overlap.
  • We can say that there does not appear to be a significant difference between the proportion of Asian adults who say that their families would welcome a white person into their families and the proportion of Asian adults who say that their families would welcome a Latino person into their families.
  • We can say that there is a significant difference between the proportion of Asian adults who say that their families would welcome a white person into their families and the proportion of Asian adults who say that their families would welcome a black person into their families.

Stanford University conducted a study of whether running is healthy for men and women over age 50. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. We are interested in the proportion of people over 50 who ran and died in the same eight-year period.

  • Explain what a “97% confidence interval” means for this study.

A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine . One of the questions asked was “What is the main problem facing the country?” Twenty percent answered “crime.” We are interested in the population proportion of adult Americans who feel that crime is the main problem.

Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem.

Refer to [link] . Another question in the poll was “[How much are] you worried about the quality of education in our schools?” Sixty-three percent responded “a lot”. We are interested in the population proportion of adult Americans who are worried a lot about the quality of education in our schools.

Construct a 95% confidence interval for the population proportion of adult Americans who are worried a lot about the quality of education in our schools.

Use the following information to answer the next three exercises: According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that “education and our schools” is one of the top issues facing California. We wish to construct a 90% confidence interval for the true proportion of California adults who feel that education and the schools is one of the top issues facing California.

A point estimate for the true population proportion is:

A 90% confidence interval for the population proportion is _______.

  • (0.761, 0.820)
  • (0.125, 0.188)
  • (0.755, 0.826)
  • (0.130, 0.183)

The error bound is approximately _____.

Use the following information to answer the next two exercises: Five hundred and eleven (511) homes in a certain southern California community are randomly surveyed to determine if they meet minimal earthquake preparedness recommendations. One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness, and 338 did not.

Find the confidence interval at the 90% Confidence Level for the true population proportion of southern California community homes meeting at least the minimum recommendations for earthquake preparedness.

  • (0.2975, 0.3796)
  • (0.6270, 0.6959)
  • (0.3041, 0.3730)
  • (0.6204, 0.7025)

The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is ______.

On May 23, 2013, Gallup reported that of the 1,005 people surveyed, 76% of U.S. workers believe that they will continue working past retirement age. The confidence level for this study was reported at 95% with a ±3% margin of error.

  • Determine the estimated proportion from the sample.
  • Determine the sample size.
  • Identify CL and α .
  • Calculate the error bound based on the information provided.
  • Compare the error bound in part d to the margin of error reported by Gallup. Explain any differences between the values.
  • Create a confidence interval for the results of this study.
  • A reporter is covering the release of this study for a local news station. How should she explain the confidence interval to her audience?

A national survey of 1,000 adults was conducted on May 13, 2013 by Rasmussen Reports. It concluded with 95% confidence that 49% to 55% of Americans believe that big-time college sports programs corrupt the process of higher education.

  • Find the point estimate and the error bound for this confidence interval.
  • Can we (with 95% confidence) conclude that more than half of all American adults believe this?
  • Use the point estimate from part a and n = 1,000 to calculate a 75% confidence interval for the proportion of American adults that believe that major college sports programs corrupt higher education.
  • Can we (with 75% confidence) conclude that at least half of all American adults believe this?
  • p′ = [latex]\frac{\text{(0}\text{.55 + 0}\text{.49)}}{\text{2}}[/latex] = 0.52; EBP = 0.55 – 0.52 = 0.03
  • No, the confidence interval includes values less than or equal to 0.50. It is possible that less than half of the population believe this.

[latex]EBP=\left(1.150\right)\sqrt{\frac{0.52\left(0.48\right)}{1,000}}\approx 0.018[/latex]

( p ′ – EBP , p ′ + EBP ) = (0.52 – 0.018, 0.52 + 0.018) = (0.502, 0.538)

STAT TESTS A: 1-PropZinterval with x = (0.52)(1,000), n = 1,000, CL = 0.75.

Answer is (0.502, 0.538)

  • Yes – this interval does not fall less than 0.50 so we can conclude that at least half of all American adults believe that major sports programs corrupt education – but we do so with only 75% confidence.

Public Policy Polling recently conducted a survey asking adults across the U.S. about music preferences. When asked, 80 of the 571 participants admitted that they have illegally downloaded music.

  • Create a 99% confidence interval for the true proportion of American adults who have illegally downloaded music.
  • This survey was conducted through automated telephone interviews on May 6 and 7, 2013. The error bound of the survey compensates for sampling error, or natural variability among samples. List some factors that could affect the survey’s outcome that are not covered by the margin of error.
  • Without performing any calculations, describe how the confidence interval would change if the confidence level changed from 99% to 90%.

You plan to conduct a survey on your college campus to learn about the political awareness of students. You want to estimate the true proportion of college students on your campus who voted in the 2012 presidential election with 95% confidence and a margin of error no greater than five percent. How many students must you interview?

CL = 0.95 α = 1 – 0.95 = 0.05 [latex]\frac{\alpha }{2}[/latex] = 0.025 [latex]{z}_{\frac{\alpha }{2}}[/latex] = 1.96. Use p ′ = q ′ = 0.5.

[latex]n=\frac{ {z}_{\frac{\alpha }{2}}{}^{2}{p}^{\prime }{q}^{\prime }}{EB{P}^{2}}= \frac{{1.96}^{2}\left(0.5\right)\left(0.5\right)}{{0.05}^{2}}=384.16[/latex]

You need to interview at least 385 students to estimate the proportion to within 5% at 95% confidence.

In a recent Zogby International Poll, nine of 48 respondents rated the likelihood of a terrorist attack in their community as “likely” or “very likely.” Use the “plus four” method to create a 97% confidence interval for the proportion of American adults who believe that a terrorist attack in their community is likely or very likely. Explain what this confidence interval means in the context of the problem.

investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student; the major characteristics of the random variable (RV) are:

It is continuous and assumes any real values. The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution. It approaches the standard normal distribution as n get larger. There is a "family of t–distributions: each representative of the family is completely defined by the number of degrees of freedom, which is one less than the number of data.

Introductory Statistics Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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Background for CDC’s Updated Respiratory Virus Guidance

Executive summary, transitioning from an emergency state, changing risk environment, unified, practical guidance, updated guidance on steps to prevent spread when you are sick is informed by numerous factors, ongoing vigilance and action.

The 2023-2024 fall and winter virus season, four years since the start of the COVID-19 pandemic, provided ongoing evidence of the changing face of respiratory diseases. COVID-19 remains an important public health threat, but it is no longer the emergency that it once was, and its health impacts increasingly resemble those of other respiratory viral illnesses, including influenza and RSV. This reality enables CDC to provide updated guidance proportionate to the current level of risk COVID-19 poses while balancing other critical health and societal needs. Key drivers and indicators of the reduction in threat from COVID-19 include:

  • Due to the effectiveness of protective tools and high degree of population immunity, there are now fewer hospitalizations and deaths due to COVID-19 . Weekly hospital admissions for COVID-19 have decreased by more than 75% and deaths by more than 90% compared to January 2022, the peak of the initial Omicron wave. Complications like multisystem inflammatory syndrome in children (MIS-C) are now also less common, and prevalence of Long COVID also appears to be decreasing. These reductions in disease severity and death have persisted through a full respiratory virus season following the expiration of the federal Public Health Emergency for COVID-19 and its associated special measures on May 11, 2023.
  • Protective tools, like vaccines and treatments, that decrease risk of COVID-19 disease (particularly severe disease) are now widely available. COVID-19 vaccination reduces the risk of symptomatic disease and hospitalization by about 50% compared to people not up to date on vaccination. Over 95% of adults hospitalized in 2023-2024 due to COVID-19 had no record of receiving the latest vaccine. Treatment with nirmatrelvir-ritonavir (Paxlovid) in persons at high risk of severe disease has been shown to decrease risk of hospitalization by 75% and death by 60% in recent studies.
  • There is a high degree of population immunity against COVID-19. More than 98% of the U.S. population now has some degree of protective immunity against COVID-19 from vaccination, prior infection, or both.

As the threat from COVID-19 becomes more similar to that of other common respiratory viruses, CDC is issuing Respiratory Virus Guidance, rather than additional virus-specific guidance. This brings a unified, practical approach to addressing risk from a range of common respiratory viral illnesses, such as influenza and RSV, that have similar routes of transmission and symptoms and similar prevention strategies. The updated guidance on steps to prevent spread when you are sick particularly reflects the key reality that many people with respiratory virus symptoms do not know the specific virus they are infected with. Importantly, states and countries that have already shortened recommended isolation times have not seen increased hospitalizations or deaths related to COVID-19. Although increasingly similar to other respiratory viruses, some differences remain, such as the risk of post-COVID conditions.

CDC will continue to evaluate available evidence to ensure the recommendations in the guidance provide the intended protection. This includes monitoring data to identify and model patterns in respiratory virus transmission, severity, hospitalizations, deaths, virus evolution, and Long COVID. In addition, CDC continues to make systems-level investments to protect the American public. Examples include measuring and enhancing effectiveness and uptake of vaccines and antiviral treatments, particularly for those at increased risk for severe disease; integrating healthcare and public health systems to prevent, identify, and respond to emerging public health threats more rapidly; and strengthening partnerships across sectors to ensure a strong public health infrastructure.

  • The Respiratory Virus Guidance covers most common respiratory viral illnesses but should not supplant specific guidance for pathogens that require special containment measures, such as measles. However, the recommendations in this guidance may still help reduce spread of various other types of infections. The guidance may not apply in certain outbreak situations when more specific guidance may be needed.
  • CDC offers separate, specific guidance for healthcare settings ( COVID-19 , flu , and general infection prevention and control )

Since 2020, CDC provided guidance specific to COVID-19, initially with detailed recommendations on many issues and for specific settings. Throughout 2022 and 2023 , CDC revised COVID-19 public health recommendations as the pandemic evolved. These changes to the guidance reflected the latest scientific evidence as well as the progression through the pandemic. The expiration of the federal Public Health Emergency for COVID-19 in 2023 also reflected a shift away from the emergency response phase  to the recovery and maintenance phases in which COVID-19 is addressed amidst many other public health threats. Measures appropriate to an emergency setting are less relevant after the emergency ended.

The continuum of pandemic phases

3a

In developing this updated Respiratory Virus Guidance, CDC carefully considered the changing risk environment, particularly lower rates of severe disease from COVID-19 and increased population immunity, as well as improvements in other prevention and control strategies.

Trends in outcomes

Hospitalizations.

In 2024, COVID-19 is less likely to result in severe disease than earlier in the pandemic because of greater immunity from vaccines and previous infections and greater treatment availability.

COVID-19 remains a greater cause of severe illness and death than other respiratory viruses, but the differences between these rates are much smaller than they were earlier in the pandemic. This difference is even smaller among people admitted to the hospital. Studies show the proportion of adults hospitalized with COVID-19 (15.5%) or influenza (13.3%) who were subsequently admitted to the intensive care unit (ICU) was similar, and patients 60 years and older hospitalized with RSV were 1.5 times more likely  to be admitted to the ICU than those with COVID-19.

Hospital admissions for COVID-19 peaked in January 2022 with more than 150,000 admissions per week, based on data from CDC’s National Healthcare Safety Network (NHSN) covering all U.S. hospitals. During the week ending February 17, 2023, there were 18,977 hospital admissions for COVID-19. During this same week, there were 10,480  hospital admissions for influenza.

Another data source—called Respiratory Virus Hospitalization Surveillance Network ( RESP-NET ) — collects hospitalization data on 8–10% of the U.S. population since before pandemic. The RESP-NET figures below demonstrate how COVID-19 hospitalizations have decreased over time and are now in the range of those for influenza and RSV.

COVID-19 hospitalizations have been declining year-over-year since 2022, with winter peaks more closely resembling those of influenza

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Data on weekly new hospital admissions of patients with COVID-19, influenza, and RSV from surveillance sites in the Respiratory Virus Hospitalization Surveillance Network ( RESP-NET ), which cover 8–10% of the U.S. population, October 2019–February 2024.

Cumulative annual rates of COVID-19-associated hospitalizations (Oct. 1–Sept. 30 to align with start of typical fall and winter virus season) have declined since 2021–2022

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*Hospitalization rates per 100,000 population. Data from the Respiratory Virus Hospitalization Surveillance Network (RESP-NET)  are preliminary and subject to change as more data become available. As data are received each week, prior case counts and rates are updated accordingly. Hospitalizations rates are likely to be underestimated as some hospitalizations might be missed because of undertesting, differing provider or facility testing practices, and diagnostic test sensitivity. Rates presented do not adjust for testing practices, which may differ by pathogen, age, race and ethnicity, and other demographic criteria. Surveillance for each pathogen was not conducted during the same time periods each season. For all seasons displayed, all three platforms conduct surveillance between October 1 and April 30 of each year. Surveillance for influenza hospitalizations was extended to June 11, 2022, for the 2021–2022 season, but otherwise occurred October through April each season. Surveillance for RSV hospitalizations occurred from October 2019 through April 2020. Since October 2020, surveillance for RSV hospitalizations has occurred year-round excluding May–June 2022. Surveillance for COVID-19 hospitalizations has occurred year-round since March 2020. Cumulative rates for the 2023–2024 season are not presented as surveillance is ongoing .

Hospitalizations by age group . Over time, rates of hospitalization for COVID-19 have decreased across all ages but have remained higher among adults ages ≥65 years relative to younger adults, children, and adolescents. Among older children, rates have decreased, and rates among children are now highest among infants ages <6 months. As of the end of December 2023, about 70% of hospitalizations were among people ages ≥65 years, and 14% were among those ages 50–64 years.

Increasing proportion of COVID-19 hospitalizations are in older adults, as well as the youngest children

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Percents of weekly COVID-19-associated hospitalizations, by age group — COVID-NET, March 1, 2020–January 27, 2024

The proportion of hospitalizations caused by each of these viruses in the 2022–2023 season varied by age group. Among children <5 years RSV caused the most hospitalizations. Among children and adolescents 5–17 years, influenza caused the most hospitalizations, and hospitalizations overall were the lowest in this age group. Among adults, COVID-19-associated hospitalizations were higher than those for influenza or RSV. These patterns have been broadly consistent thus far in the 2023-2024 season.

Most COVID-19, influenza, and RSV hospitalizations are in older adults and young children, with COVID-19 highest among older adults and RSV highest among young children (note differences in y-axis rates of hospitalizations across age groups)

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Weekly rates of laboratory-confirmed respiratory virus-associated hospitalizations by age, 2022–2023 season. Seasonal influenza surveillance ended in April 2023 for the 2022–2023 seasons; surveillance for COVID-19 and RSV hospitalizations was year-round. Data from RESP-NET .

As of February 10, 2024, 1,178,527 deaths from COVID-19 have been reported in the United States. In 2021 COVID-19 was the third leading cause of death (12% of all deaths) and in 2022 it was the fourth leading cause of death (5.7% of all deaths), following heart disease, cancer, and unintentional injury. In preliminary data from 2023 , COVID-19 was the 10 th leading cause of death. Among deaths from COVID-19 occurring from January–September 2023, 88% were among people ≥65 years.

COVID-19 is increasingly a contributing rather than the primary (underlying) cause of death. In 2020, COVID-19 was listed as the primary cause for 91% of deaths involving COVID-19. During January–September 2023, that number had fallen to 69%.

Reported deaths involving COVID-19 are several-fold greater than those reported to involve influenza and RSV. However, influenza and likely RSV are often underreported as causes of death. CDC estimates that from October 1, 2023, to February 17, 2024, 17,000–50,000 influenza deaths occurred, several times greater than the number of reported deaths. As such, the data on reported deaths should be interpreted with caution when assessing the true burden of deaths and comparing across diseases.

Current estimates of total COVID-19 deaths are not available, but COVID-19 deaths are not likely to be as underreported as are deaths involving influenza because of widespread COVID-19 testing and intensive focus on COVID-19 during the pandemic. Total COVID-19 deaths, accounting for underreporting, are likely to be higher than, but of the same order of magnitude as, total influenza deaths. Supporting this idea, the cumulative rate of COVID-19-associated hospitalizations during October 1, 2023–February 3, 2024, was 97 per 100,000 population, compared with 52 per 100,000 influenza-associated hospitalizations and 44 per 100,000 RSV hospitalizations. In-hospital death was about 1.8 times higher for COVID-19-associated hospitalizations (4.6%) vs. those for influenza (2.6%).

COVID-19-associated deaths based on reports on death certificates declined over 5-fold since their peak in 2020-2021 and are now at the same order of magnitude as estimated influenza deaths

*Reported death data from CDC’s National Vital Statistics System based on death certificates, available at CDC WONDER . Data from 2022-2024 are provisional and subject to change .

****Estimated influenza deaths, accounting for underreporting, based CDC modeling available here: Disease Burden of Flu , including confidence intervals. It has been long recognized that only counting deaths where influenza was recorded on death certificates would  underestimate influenza’s overall impact on mortality . Influenza can lead to death from other causes, such as pneumonia and congestive heart failure; however, it may not be listed on the death certificate as a contributing cause for multiple reasons, including a lack of testing. Therefore, CDC has an established history of using models to  estimate influenza-associated death totals . While under-reporting of deaths attributed to RSV and COVID-19 likely also occurs, regularly updated model estimates are currently not available. Modeled burden estimates for influenza are not directly comparable to death certificate derived counts for COVID-19 and RSV .

Divergence between infection and severe disease

Although hospitalizations and deaths involving COVID-19 have declined substantially since 2022, rates of infections with the virus have not. For example, the percentage of SARS-CoV-2 tests that are positive, a key indicator of community spread, reached peak levels of 14.6% in August 2023 and 12.9% in January 2024, similar to the peak levels observed in earlier years. Differences in testing practices between time periods might influence these data, but these high levels of test positivity are consistent with high levels also seen in wastewater.

SARS-CoV-2 test positivity (orange line) has remained elevated, a marker of ongoing COVID-19 spread, but deaths (blue bars) have declined substantially

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Provisional COVID-19 Deaths and COVID-19 Nucleic Acid Amplification Test (NAAT) Percent Positivity, by Week, in The United States, Reported to CDC. Sources:  Provisional Deaths from the CDC’s National Center for Health Statistics (NCHS) National Vital Statistics System (NVSS) National Respiratory and Enteric Virus Surveillance System (NREVSS) Figure from CDC’s COVID Data Tracker .

Wastewater viral activity levels of SARS-CoV-2 demonstrate ongoing community transmission

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Figure from CDC’s National Wastewater Surveillance System

Multisystem inflammatory syndrome in children (MIS-C)

MIS-C is a rare but serious condition associated with COVID-19 in which different parts of the body become inflamed. As of January 2024, more than 9,600 cases of MIS-C have been reported to CDC, including 79 children who died. Before March 2022, the end of the initial Omicron wave, most weekly totals of MIS-C cases exceeded 50, with some weeks involving >150 cases. The number of cases declined substantially after that point, with no week exceeding 25 cases. The reduction in MIS-C cases is likely due to multiple factors, including an increase in population immunity from both infection and vaccination, as well as differences in development of MIS-C associated with SARS-CoV-2 variants.

Weekly U.S. MIS-C cases (blue bars) have declined markedly despite ongoing high levels of COVID-19 test positivity (orange line)

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Long COVID or Post-COVID Conditions

CDC broadly defines Long COVID as signs, symptoms, and conditions that continue or develop ≥4 weeks after COVID-19. It can include a wide range of health conditions that can last weeks, months, or years. Although COVID-19 is becoming more similar to influenza and RSV in terms of hospitalizations and death over time, important differences remain, like the potential for these post-infection conditions. Long COVID occurs more often in people who had severe COVID-19 illness but can occur in anyone who has been infected with SARS-CoV-2, including children and people who were asymptomatic. Estimates of Long COVID vary widely and can differ based on study methods and how long after infection symptoms were assessed. Based on the nationally representative 2022 National Health Interview Survey, 3.4% of adults reported Long COVID and 0.5% of children . In Census Bureau’s Household Pulse Surveys, one quarter of people currently reporting Long COVID reported significant activity limitations.

Accumulating evidence suggests that vaccination prior to infection can reduce the risk of Long COVID . There is mixed evidence on whether the use of antivirals, including nirmatrelvir-ritonavir (Paxlovid), during the time of acute infection can reduce the risk of Long COVID. Decreases in Long COVID prevalence have been reported in several countries including the United States , United Kingdom , and Germany , likely due to less severe illness from COVID-19 overall, protection from vaccines, and possible changes in risk with new variants.

Increase in population immunity against COVID-19

Now, more than ever before, most people have some degree of protection because of underlying immunity. Data from a national longitudinal cohort of blood donors aged ≥16 years provide insight on the proportion of the population with antibodies against SARS-CoV-2 from infection, vaccination, or both (referred to as hybrid immunity). Hybrid immunity has been described as providing better protection with longer durability against severe illness compared to immunity from vaccination or infection alone.

In January 2021 , only an estimated 22% of people aged ≥16 years had antibodies against COVID-19. By the third quarter of 2023 (July–September), 98% had antibodies against SARS-CoV-2, with 14% from vaccination alone, 26% from infection alone, and 58% from both. An estimated 96% of children aged 6 months to 17 years had antibodies against SARS-CoV-2 in November–December 2022, including 92% with antibodies from a prior infection, according to blood samples from commercial laboratories. Although immunity against SARS-CoV-2 tends to decline from high levels initially generated by vaccination and infection, substantial protection persists for much longer , especially against the most severe outcomes like requiring a ventilator and death. New data show that the 2023–2024 updated COVID-19 vaccine can provide an additional layer of protection against severe disease.

Prevalence of vaccine-induced and infection-induced antibodies against SARS-CoV-2 among a cohort of U.S. blood donors ≥16 years

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Immunizations

As of February 3, 2024, 22% of adults reported they had received an updated 2023-2024 vaccine, including 42% of people aged ≥65 years. Vaccine uptake varies geographically and by other demographics. As of February 11, 2024, 40% of nursing home residents were up to date with a COVID-19 vaccine.

Reductions in COVID-19-associated hospitalizations over time could be even greater if more people, especially those at greater risk, receive updated COVID-19 vaccines. Among adults with COVID-19-associated hospitalizations during October–November 2023, over 95% had not received an updated (2023 –2024) COVID-19 vaccine , and most (70%) had also not received an updated vaccine from the previous year (2022–2023) .

Over 95% of adults hospitalized with COVID-19 during October–November 2023 had not received an updated (2023–2024) COVID-19 vaccine (Preliminary)

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Data from COVID-NET. Data are preliminary as they only include two months of hospitalization data for which the updated monovalent vaccine dose was recommended. Continued examinations of vaccine registry data are ongoing. No record of bivalent or updated monovalent dose : No recorded doses of COVID-19 bivalent or updated 2023-2024 monovalent dose. Bivalent booster, but no updated monovalent doses : Received COVID-19 bivalent booster vaccination but no record of receiving updated 2023-2024 monovalent booster dose. Updated monovalent dose: Received updated 2023-2024 monovalent dose. Persons with unknown vaccination status are excluded.

Vaccine effectiveness data provide the best real-world information on impact of COVID-19 vaccines on hospitalization. Data shown below from two studies presented at the Feb. 28–29, 2024, meeting of the Advisory Committee on Immunization Practices demonstrate that the 2023–2024 COVID-19 vaccine is associated with an additional ~50% increase in protection against COVID-19-associated hospitalization.

Vaccine Effectiveness of 2023-2024 vaccine against hospitalization among immunocompetent adults aged ≥18 years

Vaccine Effectiveness of 2023-2024 vaccine against hospitalization among immunocompetent adults aged ≥18 years

VE estimates adjusted for age, sex, race and ethnicity, geographic region, and calendar time. MMWR February 29, 2024

Data from another study suggest that these vaccines provide similar protection against disease caused by different co-circulating variants . Vaccines continue to provide protection to both people who have had a prior infection and those who have not. To be optimally protected against COVID-19, everyone 6 months and older should receive the latest CDC-recommended vaccine.

Infants <6 months are not eligible for COVID-19 vaccines but vaccination during pregnancy helps protect both pregnant people and their young infants from hospitalization due to COVID-19. For people with immunocompromising conditions , vaccine responses can be impaired, but vaccines provide protection against severe illness in this population. People who are moderately or severely immunocompromised are recommended to receive at least 1 dose of updated 2023–2024 COVID-19 vaccine.

Immunizations are the cornerstone of protection not just for COVID-19 but also for influenza. New in the 2023-2024 season, immunizations are available to protect those at highest risk from RSV, including older adults and infants.

Vaccines substantially reduce the risk of hospitalization, and many people at higher risk of severe disease are missing this layer of protection

*Data on COVID-19 vaccine effectiveness among adults for updated (2023-2024) COVID-19 vaccine and 2023-2024 seasonal influenza vaccine

**Data on 2023-2024 updated COVID-19 vaccines and 2023-2024 seasonal influenza vaccine from CDC’s National Immunization Survey (NIS) as of February 16, 2024. More detail, including confidence intervals around these point estimates, is available on CDC’s Respiratory Virus Data Channel . Data on percentage of older adults vaccinated for COVID-19 and influenza are for those 65+ years and for those 60+ years for RSV .

***RSV vaccination is recommended for older adults aged 60+ years based on shared clinical decision-making with a healthcare provider. RSV protection for young children is available through vaccination of pregnant people or use of an immunization called nirsevimab for young children. As of January 2024, an estimated 16% of pregnant people 32+ weeks gestation reported receiving RSV vaccine, and among females with an infant <8 months, 41% reported their infant received nirsevimab .

SARS-CoV-2 evolution, variants, and vaccines

RNA viruses like influenza and SARS-CoV-2, which causes COVID-19, accumulate random mutations over time as they replicate. Out of the many mutations that happen, a small number can provide advantages that lead to new variant lineages with increased fitness (e.g., infect people more easily or be more transmissible). Early in the pandemic, circulating SARS-CoV-2 genomes were relatively stable . Because the virus was so new, our immune systems did not recognize it, and the virus did not need new mutations to escape existing immunity to continue spreading. As population immunity increased and more people developed antibodies against SARS-CoV-2, this immune pressure selected for mutations that helped the virus escape from neutralizing antibodies, generating new variants. This evolving situation led to viruses that had many changes in the virus spike protein, such as early Omicron variants (e.g., BA.1, BA.2). These ongoing changes in the spike protein, called antigenic drift, from early virus lineages like the Alpha variant to the first Omicron variants resulted in significant escape from neutralizing antibodies, allowing reinfections of people who had been infected by early variants and leading to reduced vaccine effectiveness . Currently, all SARS-CoV-2 viruses circulating are descendants of the early Omicron variants.

Changes in the spike protein that enable escape from neutralizing antibodies are the major driver of SARS-CoV-2 evolution, since they allow the virus to better escape people’s existing immunity. To better target the changing virus and increase protection against new variants, the COVID-19 vaccine is periodically updated. For example, the updated COVID-19 vaccine for 2023–2024 includes uses XBB.1.5 antigen, a variant that was dominant for much of 2023.

A wide range of SARS-CoV-2 variants have been causing infections over time, most recently dominated by JN.1, representing increased transmission or immune escape by successive variants

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Figure based on CDC genomic surveillance data

In 2023, a variant called BA.2.86 emerged with many changes in the spike protein compared to other circulating variants, raising concerns that it might lead to a similar degree of immune escape as the initial Omicron variant. This variant, in the form of its offspring JN.1—just one mutation different from BA.2.86, displaced the other co-circulating SARS-CoV-2 variants, demonstrating it had higher fitness than other variants. However, vaccines continued to work well against JN.1 , and the number of U.S. COVID-19-associated hospitalizations occurring at this time did not exceed that of the previous year. These findings suggest that hybrid immunity induced by the updated vaccines, provided robust cross-protection against this variant and likely a wide range of variants, although continued vigilance is critical.

SARS-CoV-2 will continue to evolve, and new variants will continue to replace previous viruses. Therefore, genomic surveillance is used to identify and track variants , and representative viruses are phenotypically characterized as part of coordinated global efforts to develop updated vaccines as needed . CDC along with partners (e.g., National Institutes of Health, Food and Drug Administration, Biomedical Advanced Research and Development Authority, and World Health Organization) continue to conduct genetic surveillance to monitor for new variants, perform epidemiologic and laboratory studies to understand immune escape, and monitor key indicators like hospitalizations and emergency department visits to help inform prevention strategies. This is a continuous and iterative process that will help prepare for the upcoming 2024–2025 fall and winter season.

SARS-CoV-2 shedding and transmission dynamics

Even as the SARS-CoV-2 virus has continued to evolve, the duration of shedding infectious virus has remained relatively consistent, with most individuals no longer infectious after 8-10 days. The presence of certain COVID-19 symptoms, most prominently fever, is associated with greater infectious virus on the day of symptom. The highest levels of culturable virus typically occur within a few days before and after symptom onset. Since Omicron BA.1 variant, there is a slightly shorter time between infection to symptom onset than previous variants. Overall, these data suggest most SARS-CoV-2 transmission, regardless of variant, largely occurs early in the course of illness.

Notably, over half of SARS-CoV-2 community transmission is estimated to come from people who are asymptomatic at the time, including both pre-symptomatic and asymptomatic individuals, meaning exposure to the virus in the community from people who do not know they are infected is likely common.

Highest levels of culture-positive SARS-CoV-2, an indicator of infectiousness, occur in the days around and after symptom onset, with a small proportion of people continuing to have culturable virus beyond one week

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Unpublished data from the Respiratory Virus Transmission Network , involving five U.S. sites that enrolled people who tested positive for SARS-CoV-2 and their household contacts during November 2022–May 2023. Onset was defined as first day of symptoms or, if asymptomatic, first positive test. Note that people can have positive PCR tests, which detect viral genetic material, after they are no longer shedding infectious virus, and culture is the best indicator of infectious virus. This figure is similar to one previously published based on data from early 2021, underscoring the overall stability of viral shedding across variants.

In the Delta variant era, vaccination was associated with reduced infectious virus, demonstrating the potential impact of immunity on viral shedding and transmission. Immunity from vaccination, as well as previous infections, wanes over time, which likely attenuates this impact. Additionally, the continued evolution of variants better able to escape existing immunity may also affect the impact of vaccination and previous infection on shedding of infectious virus.

Improvements related to other prevention and control strategies

In addition to greater population understanding of effective prevention strategies like hand hygiene, respiratory etiquette, cleaning, masks, and physical distancing, advancements in awareness, accessibility, and the science base related to treatment, air quality, tests, and steps to prevent spread when you’re sick have also enabled people to act to lower the risk from respiratory viruses.

Several medications are available for outpatient treatment of mild to moderate COVID-19 for people at increased risk of severe illness. Data for nirmatrelvir-ritonavir (Paxlovid), the first-line drug available for oral use, suggest that it reduce the risk of hospitalization and death by half or more . For example, a systematic review and meta-analysis of studies that examined nirmatrelvir-ritonavir effectiveness and efficacy found that people who received nirmatrelvir-ritonavir had 75% lower odds of death and 60% lower odds of hospitalization. People who received nirmatrelvir-ritonavir had 83% lower odds of hospitalization and death as a composite outcome compared with people who did not use nirmatrelvir-ritonavir.

However, uptake of these treatments remains suboptimal , meaning many people are missing this layer of protection against hospitalization and death. A study of patients in the Veterans Health Administration reported that among all persons with SARS CoV-2 infection, 24% used outpatient antiviral medications in 2022, remaining at that level through early 2023. Similar overall rates of use, with a maximum of 34%, were found using observational data of a large cohort from health care systems participating in the National Patient-Centered Clinical Research Network (PCORnet). This study also highlighted racial and ethnic differences in treatment uptake. During April–July 2022, treatment with nirmatrelvir-ritonavir among adults aged ≥20 years was 35.8%, 24.9%, 23.1%, and 19.4% lower among Black, multiple or other race, American Indian or Alaska native or other Pacific Islander, and Asian patients, respectively, than among white patients (31.9% treated). A CDC study found that among 699,848 U.S. adults aged ≥18 years eligible for nirmatrelvir-ritonavir during April–August 2022, 28.4% received a prescription with 5 days of being diagnosed with COVID-19.

CDC and NIH continue to monitor real-world effectiveness data for COVID-19 treatment. Current evidence suggests that effectiveness of nirmatrelvir-ritonavir is retained among persons who have been vaccinated and confers incremental benefit among persons at high risk for severe disease, although this is an underutilized treatment.

Air quality

Ventilation and related strategies to improve indoor air quality can reduce infective viral particle concentrations in indoor air. In 2023, informed by accumulating evidence, CDC issued recommendations for using Minimum Efficiency Reporting Value (MERV) 13 or greater and getting at least 5 air changes per hour of clean air in occupied spaces through air flow, filtration, or air treatment. CDC’s Interactive Home Ventilation Tool can help people identify strategies they can use to decrease the level of viral particles in their home. CDC also now provides a similar tool for building owners and operators. In addition, the U.S. Government issued a Clean Air in Buildings Challenge to help building owners and operators improve indoor air quality and protect public health.

Laboratory tests are currently widely available and can be readily accessed for diagnosis of COVID-19, influenza, and RSV. At-home antigen tests for SARS-CoV-2 are also widely available and increasingly familiar to the public. At-home rapid tests for influenza have recently received FDA approval and may become more widely accessible over time.

Staying home when sick and other steps to prevent spread

The importance of staying home and away from others when sick became more widely understood during the COVID-19 pandemic. When individuals have the option to stay home and be compensated while sick , they are much more likely to do so. Similarly, people with prior telework experience are more apt to work from home when they have respiratory symptoms , rather than work in person at an office.

Unlike early in the pandemic when COVID-19 was nearly the only respiratory virus causing illness, it is now one of many, including influenza , RSV , adenoviruses , rhinoviruses , enteroviruses , human metapneumovirus , parainfluenza virus , and other common human coronaviruses . CDC is focusing guidance on the core measures that provide the most protection across respiratory viruses. The updated guidance emphasizes the importance of staying home and away from others when sick from respiratory viruses, regardless of the virus, as well as additional preventive actions.

Virus not known in most respiratory infections

Viruses cause most acute respiratory illnesses, but it is rarely possible to determine the type of virus without testing, and oftentimes testing does not change clinical management. Testing for most respiratory pathogens is rarely available outside of healthcare settings. Although at-home antigen testing is widely available for COVID-19, most infections likely go undiagnosed. In a recent CDC survey, less than half of people said they would do an at-home test for COVID-19 if they had cold or cough symptoms, and less than 10% said they would get tested at a pharmacy or by a healthcare provider.

Even when testing occurs, COVID-19 is often not identified early in illness. The overall sensitivity of COVID-19 antigen tests is relatively low and even lower in individuals with only mild symptoms. Significant numbers of false negative test results occur early in an infection. This means mildly symptomatic cases are not always detected, and when they are detected, it often occurs several days into an illness, which is typically past when peak infectiousness occurs.

Public interest in prevention is not limited to COVID-19

A November 2023 survey from the Harvard Opinion Research Program found people were not meaningfully more concerned about any one respiratory virus, with roughly similar proportions reporting being concerned about getting infected with COVID-19, seasonal influenza, RSV, and a cold. Relatedly, a CDC survey found that a majority of Americans take precautions when sick with cold or cough symptoms (i.e., avoiding contact with people at higher risk, avoiding large indoor gatherings) even if they don’t know what virus is causing the illness.

Respiratory Virus Guidance does not imply all viruses are the same

Respiratory viruses are certainly not all the same. Some, like SARS-CoV-2, spread more through respiratory particles in the air, whereas others, like RSV and adenovirus , are thought to also spread via surface transmission. As such, this guidance is not meant to apply to specialized situations, like healthcare or certain disease outbreaks , in which more detailed guidance specific to the pathogen may be warranted. For example, adenoviruses are resistant to many common disinfectants and can remain infectious for hours on environmental surfaces. For the general public, however, an overall focus on hygiene, indoor air improvements, and mask use, coupled with necessarily specific recommendations about vaccines and treatment, provides a practical approach that addresses the key prevention measures.

The updated Respiratory Virus Guidance recommends people with respiratory virus symptoms that are not better explained by another cause stay home and away from others until at least 24 hours after both resolution of fever AND overall symptom are getting better. This recommendation addresses the period of greatest infectiousness and highest viral load for most people, which is typically in the first few days of illness and when symptoms, including fever, are worst. This is similar to longstanding recommendations for other respiratory illnesses, including influenza.

A residual risk of SARS-CoV-2 transmission remains , depending on the person and circumstances, after the period in which people are recommended to stay home and away from others. Five additional days of interventions (i.e., masking, testing, distancing, improved air quality, hygiene, and/or testing) reduce harm during later stages of illness, especially to protect people at higher risk of severe illness. Some people, especially people with weakened immune systems, might be able to infect others for an even longer time. It is important to note that a similar residual risk of transmission is also true for influenza and other viruses.

In addition to the overall reduction in risk from COVID-19, other factors considered in developing this component of the guidance included assessment of personal and societal costs of extended isolation (e.g., limited paid sick time), analysis of the period of peak infectiousness ( see section 4.), and acknowledgement that many people with respiratory virus symptoms do not often know the pathogen that is causing their illness.

Case examples from states and countries that changed their COVID-19 isolation guidance to recommendations similar to CDC’s updated guidance did not experience clear increases in community transmission or hospitalization rates. Examples include the most populous Canadian provinces ( Ontario , Quebec , and British Columbia ), Australia , Denmark , France , and Norway , as well as California (on January 9, 2024) and Oregon (May 2023). In California and Oregon, for the week ending February 10, COVID-19 test positivity, emergency department visits, and hospitalizations were lower than the national average.

No appreciable difference in COVID-19 ED and hospitalization trends in Oregon vs. nation or neighboring Washington after guidance change

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Data from CDC’s COVID Data Tracker

Need for ongoing implementation of recommendations

Vaccines remain an underused layer of protection, even for groups at higher risk. For example, only 42% of adults aged 65 years or greater had received an updated COVID-19 vaccine as of February 16, 2024, compared with 73% for flu . COVID-19 antiviral treatments are also substantially underused to prevent severe COVID-19, meaning many people are missing out on important protection. Influenza treatment is also underused.

Ongoing data monitoring

The SARS-CoV-2 virus will continue to evolve, and new variants will continue to replace previous viruses. Genomic surveillance to monitor for new variants, epidemiologic studies to understand immune escape, infectiousness, severity, and monitoring of key indicators like hospitalizations and emergency department visits, all help inform prevention strategies.

Various data systems are in place to continue to monitor for changes in how COVID-19 affects us . These include monitoring laboratory-based percent positivity and wastewater as indicators of changes in infections. Data on hospitalizations and deaths are indicators of severe illness while data on hospital occupancy and capacity provide information on stress on the healthcare system. Epidemiologic studies continue to assess how infectious the virus is and how efficiently it transmits between people as well as the severity of disease it causes. Ongoing monitoring through genomic surveillance and viral characterization will continue to be important to identify and describe new SARS-CoV-2 variants that may emerge. Vaccines will continue to be updated based on circulating variants, and other protective measures can be scaled up as needed. If variants emerge that have significant immune escape from existing vaccines and therapeutics, non-pharmaceutical interventions such as masking, distancing, and ventilation will be particularly important.

COVID-19 remains an important public health threat, but it is no longer the emergency that it once was, and its health impacts increasingly resemble those of other respiratory viral illnesses, including influenza and RSV.

Protective tools, like vaccination and treatment that decrease risks of COVID-19 disease are now widely available and resultantly, far fewer people are getting seriously ill from COVID-19. Complications like multisystem inflammatory syndrome in children (MIS-C) and Long COVID are now less common as well. Data indicate rates of hospitalizations and deaths are down substantially, and that clinically COVID-19 has become similar to, or even less severe in hospitalized people, than influenza and RSV.

These factors have enabled CDC to issue updated Respiratory Virus Guidance that provides the public with recommendations and information about effective steps and strategies tailored to the current level of risk posed by COVID-19 and other common respiratory viral illnesses.

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Primary election results: Congressional and California

March 5, 2024

Deadlines and results

The polls have closed in California. All mailed ballots need to be postmarked on or before that date. Mail-in, provisional and conditional ballots will be accepted, processed and counted for several days after election day. The data on this page will update periodically until all results are in.

We’re tracking races across California, including primary elections for Democratic and Republican presidential nominees . Results for a statewide proposition , U.S. Senate and House seats, and state Senate and Assembly contests are also available on this page.

In state-level primary races, the top two finishers will move on to the general election in November. Their names will be indicated with checkmarks once their races are called by the Associated Press.

Initial results are expected shortly after the polls close at 8 p.m.

Every registered voter in the state receives a ballot by mail. To vote by mail, these ballots must be postmarked by March 5. They may take several days to process. Results from provisional and conditional ballots also take longer, and will be added to the tally once they are cleared.

The data on this page updates periodically as results come in from the Associated Press.

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Presidential races

California is one of 16 states and one U.S. territory holding their 2024 presidential primaries on Super Tuesday. Both major parties have clear front-runners, but their nominations are not secure until a candidate wins a majority of pledged delegates. Read more about the national election here .

More than a third of total delegates for president are awarded on this busy primary day. At stake are 1,421 Democratic delegates (424 from California) and 874 Republican delegates (169 from California).

✓ Winner * Incumbent

Republicans

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Congressional races

U.s. senate.

This is the state’s first U.S. Senate election without an incumbent since 2016, and 27 people are on the ballot , seeking to finish in the top two on March 5 and move on to the general election. But Californians actually have two Senate races to vote on.

In September, Gov. Gavin Newsom appointed Laphonza Butler to take the late Sen. Dianne Feinstein’s seat until a replacement could be elected. Whoever wins this race will serve two months, from the general election on Nov. 5 through the end of the term on Jan. 3, 2025.

For remainder of term ending Jan. 3, 2025

The other Senate race on the primary ballot is more crucial: It will determine who serves the next full six-year term as California’s junior senator, from January 2025 through January 2031.

For a six-year term ending January 2031

A few competitive congressional districts in California could change GOP prospects for the general election. The San Joaquin Valley’s District 22 and Southland districts including 40, 45 and 47 are considered potential swing seats.

House seats in three districts in the Los Angeles area – represented by outgoing Democratic Reps.

Tony Cárdenas of Pacoima, Adam Schiff of Burbank and Grace Napolitano of Norwalk – are all open.

Icon of California

Statewide races

Proposition 1.

Proposition 1 is the only state legislative proposition on the ballot this March.

The proposition asks voters to approve major changes to the state’s 20-year-old Mental Health Services Act to better serve Californians with substance-use disorders. A “yes” vote on the measure would also approve a $6.38-billion bond to build facilities to provide 10,000 new treatment beds. Read more about the measure here .

Proposition Behavioral Health Services Program

State Senate

State assembly.

  • Bringing It Together: Homework
  • Introduction
  • 1.1 Definitions of Statistics, Probability, and Key Terms
  • 1.2 Data, Sampling, and Variation in Data and Sampling
  • 1.3 Frequency, Frequency Tables, and Levels of Measurement
  • 1.4 Experimental Design and Ethics
  • 1.5 Data Collection Experiment
  • 1.6 Sampling Experiment
  • Chapter Review
  • 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
  • 2.2 Histograms, Frequency Polygons, and Time Series Graphs
  • 2.3 Measures of the Location of the Data
  • 2.4 Box Plots
  • 2.5 Measures of the Center of the Data
  • 2.6 Skewness and the Mean, Median, and Mode
  • 2.7 Measures of the Spread of the Data
  • 2.8 Descriptive Statistics
  • Formula Review
  • 3.1 Terminology
  • 3.2 Independent and Mutually Exclusive Events
  • 3.3 Two Basic Rules of Probability
  • 3.4 Contingency Tables
  • 3.5 Tree and Venn Diagrams
  • 3.6 Probability Topics
  • Bringing It Together: Practice
  • 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
  • 4.2 Mean or Expected Value and Standard Deviation
  • 4.3 Binomial Distribution
  • 4.4 Geometric Distribution
  • 4.5 Hypergeometric Distribution
  • 4.6 Poisson Distribution
  • 4.7 Discrete Distribution (Playing Card Experiment)
  • 4.8 Discrete Distribution (Dice Experiment Using Three Regular Dice)
  • 5.1 Continuous Probability Functions
  • 5.2 The Uniform Distribution
  • 5.3 The Exponential Distribution
  • 5.4 Continuous Distribution
  • 6.1 The Standard Normal Distribution
  • 6.2 Using the Normal Distribution
  • 6.3 Normal Distribution (Lap Times)
  • 6.4 Normal Distribution (Pinkie Length)
  • 7.1 The Central Limit Theorem for Sample Means (Averages)
  • 7.2 The Central Limit Theorem for Sums
  • 7.3 Using the Central Limit Theorem
  • 7.4 Central Limit Theorem (Pocket Change)
  • 7.5 Central Limit Theorem (Cookie Recipes)
  • 8.1 A Single Population Mean using the Normal Distribution
  • 8.2 A Single Population Mean using the Student t Distribution
  • 8.3 A Population Proportion
  • 8.4 Confidence Interval (Home Costs)
  • 8.5 Confidence Interval (Place of Birth)
  • 8.6 Confidence Interval (Women's Heights)
  • 9.1 Null and Alternative Hypotheses
  • 9.2 Outcomes and the Type I and Type II Errors
  • 9.3 Probability Distribution Needed for Hypothesis Testing
  • 9.4 Rare Events, the Sample, Decision and Conclusion
  • 9.5 Additional Information and Full Hypothesis Test Examples
  • 9.6 Hypothesis Testing of a Single Mean and Single Proportion
  • 10.1 Two Population Means with Unknown Standard Deviations
  • 10.2 Two Population Means with Known Standard Deviations
  • 10.3 Comparing Two Independent Population Proportions
  • 10.4 Matched or Paired Samples
  • 10.5 Hypothesis Testing for Two Means and Two Proportions
  • 11.1 Facts About the Chi-Square Distribution
  • 11.2 Goodness-of-Fit Test
  • 11.3 Test of Independence
  • 11.4 Test for Homogeneity
  • 11.5 Comparison of the Chi-Square Tests
  • 11.6 Test of a Single Variance
  • 11.7 Lab 1: Chi-Square Goodness-of-Fit
  • 11.8 Lab 2: Chi-Square Test of Independence
  • 12.1 Linear Equations
  • 12.2 Scatter Plots
  • 12.3 The Regression Equation
  • 12.4 Testing the Significance of the Correlation Coefficient
  • 12.5 Prediction
  • 12.6 Outliers
  • 12.7 Regression (Distance from School)
  • 12.8 Regression (Textbook Cost)
  • 12.9 Regression (Fuel Efficiency)
  • 13.1 One-Way ANOVA
  • 13.2 The F Distribution and the F-Ratio
  • 13.3 Facts About the F Distribution
  • 13.4 Test of Two Variances
  • 13.5 Lab: One-Way ANOVA
  • A | Review Exercises (Ch 3-13)
  • B | Practice Tests (1-4) and Final Exams
  • C | Data Sets
  • D | Group and Partner Projects
  • E | Solution Sheets
  • F | Mathematical Phrases, Symbols, and Formulas
  • G | NOTEs for the TI-83, 83+, 84, 84+ Calculators

A certain small town in the United states has a population of 27,873 people. Their ages are as follows:

  • Construct a histogram of the age distribution for this small town. The bars will not be the same width for this example. Why not? What impact does this have on the reliability of the graph?
  • What percentage of the community is under age 35?
  • Which box plot most resembles the information above?

Javier and Ercilia are supervisors at a shopping mall. Each was given the task of estimating the mean distance that shoppers live from the mall. They each randomly surveyed 100 shoppers. The samples yielded the following information.

  • How can you determine which survey was correct ?
  • Explain what the difference in the results of the surveys implies about the data.

Use the following information to answer the next three exercises : We are interested in the number of years students in a particular elementary statistics class have lived in California. The information in the following table is from the entire section.

What is the IQR ?

What is the mode?

Is this a sample or the entire population?

  • entire population

Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows:

  • Find the sample mean x ¯ x ¯ .
  • Find the approximate sample standard deviation, s .

Forty randomly selected students were asked the number of pairs of sneakers they owned. Let X = the number of pairs of sneakers owned. The results are as follows:

  • Find the sample mean x – x –
  • Find the sample standard deviation, s
  • Construct a histogram of the data.
  • Complete the columns of the chart.
  • Find the first quartile.
  • Find the median.
  • Find the third quartile.
  • Construct a box plot of the data.
  • What percent of the students owned at least five pairs?
  • Find the 40 th percentile.
  • Find the 90 th percentile.
  • Construct a line graph of the data
  • Construct a stemplot of the data

Following are the published weights (in pounds) of all of the team members of the San Francisco 49ers from a previous year.

177; 205; 210; 210; 232; 205; 185; 185; 178; 210; 206; 212; 184; 174; 185; 242; 188; 212; 215; 247; 241; 223; 220; 260; 245; 259; 278; 270; 280; 295; 275; 285; 290; 272; 273; 280; 285; 286; 200; 215; 185; 230; 250; 241; 190; 260; 250; 302; 265; 290; 276; 228; 265

  • Organize the data from smallest to largest value.
  • The middle 50% of the weights are from _______ to _______.
  • If our population were all professional football players, would the above data be a sample of weights or the population of weights? Why?
  • the population mean, μ .
  • the population standard deviation, σ .
  • the weight that is two standard deviations below the mean.
  • When Steve Young, quarterback, played football, he weighed 205 pounds. How many standard deviations above or below the mean was he?
  • That same year, the mean weight for the Dallas Cowboys was 240.08 pounds with a standard deviation of 44.38 pounds. Emmit Smith weighed in at 209 pounds. With respect to his team, who was lighter, Smith or Young? How did you determine your answer?

One hundred teachers attended a seminar on mathematical problem solving. The attitudes of a representative sample of 12 of the teachers were measured before and after the seminar. A positive number for change in attitude indicates that a teacher's attitude toward math became more positive. The 12 change scores are as follows:

3 ; 8 ; –1 ; 2 ; 0 ; 5 ; –3 ; 1 ; –1 ; 6 ; 5 ; –2

  • What is the mean change score?
  • What is the standard deviation for this population?
  • What is the median change score?
  • Find the change score that is 2.2 standard deviations below the mean.

Refer to Figure 2.51 determine which of the following are true and which are false. Explain your solution to each part in complete sentences.

  • The medians for all three graphs are the same.
  • We cannot determine if any of the means for the three graphs is different.
  • The standard deviation for graph b is larger than the standard deviation for graph a.
  • We cannot determine if any of the third quartiles for the three graphs is different.

In a recent issue of the IEEE Spectrum , 84 engineering conferences were announced. Four conferences lasted two days. Thirty-six lasted three days. Eighteen lasted four days. Nineteen lasted five days. Four lasted six days. One lasted seven days. One lasted eight days. One lasted nine days. Let X = the length (in days) of an engineering conference.

  • Organize the data in a chart.
  • Find the median, the first quartile, and the third quartile.
  • Find the 65 th percentile.
  • Find the 10 th percentile.
  • The middle 50% of the conferences last from _______ days to _______ days.
  • Calculate the sample mean of days of engineering conferences.
  • Calculate the sample standard deviation of days of engineering conferences.
  • Find the mode.
  • If you were planning an engineering conference, which would you choose as the length of the conference: mean; median; or mode? Explain why you made that choice.
  • Give two reasons why you think that three to five days seem to be popular lengths of engineering conferences.

A survey of enrollment at 35 community colleges across the United States yielded the following figures:

6414; 1550; 2109; 9350; 21828; 4300; 5944; 5722; 2825; 2044; 5481; 5200; 5853; 2750; 10012; 6357; 27000; 9414; 7681; 3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493; 2771; 2861; 1263; 7285; 28165; 5080; 11622

  • Organize the data into a chart with five intervals of equal width. Label the two columns "Enrollment" and "Frequency."
  • If you were to build a new community college, which piece of information would be more valuable: the mode or the mean?
  • Calculate the sample mean.
  • Calculate the sample standard deviation.
  • A school with an enrollment of 8000 would be how many standard deviations away from the mean?

Use the following information to answer the next two exercises. X = the number of days per week that 100 clients use a particular exercise facility.

The 80 th percentile is _____

The number that is 1.5 standard deviations BELOW the mean is approximately _____

  • Cannot be determined

Suppose that a publisher conducted a survey asking adult consumers the number of fiction paperback books they had purchased in the previous month. The results are summarized in the Table 2.84 .

  • Are there any outliers in the data? Use an appropriate numerical test involving the IQR to identify outliers, if any, and clearly state your conclusion.
  • If a data value is identified as an outlier, what should be done about it?
  • Are any data values further than two standard deviations away from the mean? In some situations, statisticians may use this criteria to identify data values that are unusual, compared to the other data values. (Note that this criteria is most appropriate to use for data that is mound-shaped and symmetric, rather than for skewed data.)
  • Do parts a and c of this problem give the same answer?
  • Examine the shape of the data. Which part, a or c, of this question gives a more appropriate result for this data?
  • Based on the shape of the data which is the most appropriate measure of center for this data: mean, median or mode?

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Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
  • Authors: Barbara Illowsky, Susan Dean
  • Publisher/website: OpenStax
  • Book title: Introductory Statistics 2e
  • Publication date: Dec 13, 2023
  • Location: Houston, Texas
  • Book URL: https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
  • Section URL: https://openstax.org/books/introductory-statistics-2e/pages/2-bringing-it-together-homework

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IMAGES

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    Section 7.2 Homework. Refer to the accompanying data display that results from a sample of airport data speeds in Mbps. (13.046,22.15) a. Express the confidence interval in the format that uses the "less than" symbol. Given that the original listed data use one decimal place, round the confidence interval limits accordingly. b.

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  3. Ch. 7 Solutions

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  24. Ch. 2 Bringing It Together: Homework

    Let X = the length (in days) of an engineering conference. Organize the data in a chart. Find the median, the first quartile, and the third quartile. Find the 65 th percentile. Find the 10 th percentile. Construct a box plot of the data. The middle 50% of the conferences last from _______ days to _______ days.

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