9. Hypothesis Testing 2

In Hypothesis Testing 1, you were introduced to the ideas of hypothesis testing in the context of deciding whether a coin was fair or biased in favor of heads.  In this section hypothesis testing concerning population means is explored.

In all three methods it is assumed that the distribution of sample means for samples of size n is, at least, approximately normal with mean given by µ0 (assume that µ0 is 50 for the next examples) from the null hypothesis, and standard deviation sigma/Sqrt[n] (which equals 5/Sqrt[36]=5/6).

Step 1: Based on alpha and the alternative hypothesis, find a critical z-value or z-values (1.645 for the example).

Step 2: Take the random sample and compute the sample mean, xbar (for example, suppose the sample xbar that you get is 52.3).

Step 3: Put xbar along with mu, sigma, and n into .   The resulting z-value is called the computed z-value. (For the example, the computed z-value is 2.76).  If the computed z-value lies outside of the critical z-value found in 1, reject the null hypothesis; otherwise, fail to reject the null hypothesis.  (For the example, 2.76 lies outside of 1.645, so you would reject the null hypothesis)

Step 1: Take a random sample and compute the sample mean, xbar.  (Suppose, for example, that xbar is 52.3)

Step 2: Put the computed xbar along with mu, sigma, and n into .  (For the example the computed z-value is 2.76)  Find the probability in the tail beyond (beyond is determined by the alternative hypothesis) the computed z-value.  If the test is a 1-tail test, this probability is called the p-value.  (For the example the p-value=0.003). If the test is a 2-tail test, double the probability to find the p-value.

Step 3: If the p-value is less than or equal to alpha, reject the null hypothesis.  If the p-value is greater than alpha, fail to reject the null hypothesis.  (For the example, since 0.003<0.05, you would reject the null hypothesis)

To review, a Type I error occurs when the null hypothesis is true but the test rejects it, while a Type II error occurs when the null hypothesis is false but the hypothesis test accepts it.  P[Type I error]=Alpha and P[Type II error]=Beta.  In the next examples Type I and Type II errors will be calculated in hypothesis tests for a population mean.

In all the following examples assume that a random sample of size 36 has been taken from a population with standard deviation 5.  Assume that the sample mean for this sample of size 36 is xbar=52.7.  Finally, assume that the significance level, alpha, is 0.05.

Using the critical xbar method, the critical z-value is 1.645.  From the equation youget 1.645=(critical xbar-50)/(5/6), and solving this for the critical xbar results in critical xbar=51.37.  Since the sample xbar is 52.7, you would reject the null hypothesis in favor of the alternative.  What is beta, the probability of a Type II error if the population mean is in fact 52?  The computation is not shown here but in the title section of the next graphyou see that beta is 0.22.

This example mirrors example 1.  Again, using the critical xbar method, the critical z-value is -1.645.  From the equation you get -1.645=(critical xbar-50)/(5/6), and solving this for the critical xbar results in critical xbar=48.63.  Since the sample xbar is 52.7, you would accept the null hypothesis.  What is beta, the probability of a Type II error if the population mean is in fact 52?  The computation is not shown but beta is equal to 0.99.

In Example 3, you the alternative hypothesis leads you to reject the null hypothesis for either large or small values of xbar.  This is a two tail test.  There are two critical z-values, -1.96 and +1.96.  From the equation you get -1.96=(critical xbar-50)/(5/6), and +1.96=(critical xbar-50)/(5/6).  Solving these two equations results in the critical xbar values of 48.37 and 51.63.  Since the sample xbar is 52.7, you would reject the null hypothesis.  What is beta, the probability of a Type II error if the population mean is in fact 52?  The computation is again not but from the title section of the next graph you see that beta is 0.33.  The red shaded area is alpha, the probability of a Type I error, and the blue shading is beta, the probability of a Type II error.

In finding confidence intervals for the population mean, for small samples from a normally distributed population where the population standard deviation was unknown, you had to use Student's t-distribution to complete the solution.  The situation is the same in hypothesis testing.  Instead of using , you must use where the expression has a t-distribution with n-1 degrees of freedom.

First, the null and alternative hypotheses must be stated.  They are

H0: µ=10 and Ha: µ<>10 where '<>' means not equal.

Since the statement of the example expresses no preference in determining whether the mean of the population is less than or greater than 10, the 'not equal' alternative hypothesis is used.  The p-value approach using the t-statistic shown above is employed. 

From the random sample values, you find xbar=8.9 and s=0.84. 

Substituting them into the t-statistic formula, you get calculated t=(8.9-10)/(0.84/Sqrt(4))=(-1.1)/(0.84/2)=-2.62. 

Since a two tail test is indicated by the alternative hypothesis, the p-value is found by computing the area under Student's t-distribution with 3 degrees of freedom to the left of -2.62 and doubling the result.  Student's t-distribution is symmetric about zero, so instead of finding the area to the left of -2.62, you can find the area to the right of 2.62.  In the row of the 't-table' corresponding to 3 degrees of freedom, you find 2.353 and 3.182.  The number that you seek, 2.62 lies between these two numbers.  Then the area under the Student's t-distribution with 3 degrees of freedom to the right of 2.62 must lie between 0.025 and 0.05, the numbers at the top of the columns corresponding to 2.353 and 3.182.  The p-value lies between 2*0.025=0.05 and 2*0.05=0.10. 

The p-value is larger than 0.05 so the null hypothesis is accepted.

2024 Polestar 2 RWD First Test: Feeling Better

Airport Traveler Testing Program for SARS-CoV-2

Travel can facilitate SARS-CoV-2 introduction. To reduce introduction of SARS-CoV-2 infections, the state of Alaska implemented a program on June 6, 2020, for arriving air, sea, and road travelers that required either molecular testing for SARS-CoV-2, the virus that causes COVID-19, or a 14-day self-quarantine after arrival. The Alaska Department of Health and Social Services (DHSS) used weekly standardized reports submitted by 10 participating Alaska airports to evaluate air traveler choices to undergo testing or self-quarantine, traveler test results, and airport personnel experiences while implementing the program. Among 386,435 air travelers who arrived in Alaska during June 6–November 14, 2020, a total of 184,438 (48%) chose to be tested within 72 hours before arrival, 111,370 (29%) chose to be tested on arrival, and 39,685 (10%) chose to self-quarantine without testing after arrival. An additional 15,112 persons received testing at airport testing sites; these were primarily travelers obtaining a second test 7–14 days after arrival, per state guidance. Of the 126,482 airport tests performed in Alaska, 951 (0.8%) results were positive, or one per 406 arriving travelers. Airport testing program administrators reported that clear communication, preparation, and organization were vital for operational success; challenges included managing travelers' expectations and ensuring that sufficient personnel and physical space were available to conduct testing. Expected mitigation measures such as vaccination, physical distancing, mask wearing, and avoidance of gatherings after arrival might also help limit postarrival transmission. Posttravel self-quarantine and testing programs might reduce travel-associated SARS-CoV-2 transmission and importation, even without enforcement. Traveler education and community and industry partnerships might help ensure success.

To assess the airport traveler testing program, Alaska DHSS reviewed Alaska's COVID-19 requirements and testing operations for arriving air travelers during June 6–November 14, 2020. Although travelers entering Alaska by road and sea were also subject to these requirements, entry by road and sea was minimal after Canada began restricting nonessential transit on March 20, 2020,[1] and these ports of entry neither provided weekly briefs nor routinely offered onsite testing; therefore, this report is limited to an analysis of the air traveler program. Airport programs were asked to provide weekly reports on the numbers of incoming flights, passengers screened for symptoms, passengers tested within 72 hours before arrival, passengers who chose to self-quarantine for 14 days after arrival, passengers tested at the airport, and positive test results. In addition to comments provided in the weekly briefs, airport program administrators from all 10 participating airports were also asked to provide improvement recommendations; five airports responded in a narrative format, from which themes were extracted. This activity was reviewed by CDC and was conducted consistent with applicable federal law and CDC policy.*

As part of the airport testing program, airports were required to screen travelers arriving from out of state for symptoms, offer testing, and record whether travelers chose testing or self-quarantine. Alaska DHSS contracted with local health organizations and enlisted local governments to staff and manage testing program operations. Program personnel collected samples within or just outside the Transportation Security Administration (TSA) secure area at all 10 airports. Specimens were analyzed by reverse transcription–polymerase chain reaction at the Alaska State Public Health Laboratories and commercial laboratories. Traveler information was initially collected on paper forms and later via the Alaska Travel Portal (i.e., COVIDSECURE), a web-based application created to manage travel-associated COVID-19 data†. The software allowed travelers to report symptoms, close contacts, and demographic information and to upload and view test results and enter their self-quarantine location.

A travel mandate implemented in Alaska during March 2020 required all travelers entering Alaska to self-quarantine for 14 days after arrival. In June, testing was introduced as an option to shorten the 14-day quarantine, with a test near the time of arrival and a second test 7–14 days after arrival. In August, the option for a 14-day self-quarantine without testing was removed for nonresidents; testing before travel was encouraged for nonresidents, who were charged a $250 fee if they chose to test at the airport on arrival. Starting in October, the requirement for a second test 7–14 days after arrival was removed (Box).

During June 6–November 14, 2020, a total of 386,435 air travelers who arrived in Alaska were screened for symptoms; 184,438 (48%) arrived with proof of a negative or pending SARS-CoV-2 test result, 111,370 (29%) chose to be tested on arrival, and 39,685 (10%) chose to self-quarantine after arrival for 14 days without testing (Figure 1). The remaining 50,942 (13%) travelers were exempt from the testing and quarantine requirements because they 1) were following an alternative workforce protection plan outlining alternative strategies to reduce the risk for importation that had been submitted by their employer to the state, 2) arrived with a previous positive test result and proof of completion of isolation, 3) had traveled outside Alaska for <72 hours, 4) left the airport before screening, or 5) were a child exempt from screening requirements because of age. Weekly airport briefs submitted to Alaska DHHS indicated that <10 travelers each week were noncompliant with registration or screening. An additional 15,112 persons received testing at airport testing sites; these were primarily travelers obtaining a second test 7–14 days after arrival, per state guidance.

Figure 1.

Number of air travelers* who chose self-quarantine after arrival or SARS-CoV-2 testing before travel or at airport on arrival,† by date§ — 10 airports, Alaska, June 6–November 14, 2020¶*Paper forms used by certain airports before August 15, 2020, allowed some travelers to select multiple options.†The travel mandate required two tests (one near the time of arrival and a second test 7–14 days after arrival); the first test date for those tested in the airport is shown, calculated by subtracting the number of second-test vouchers redeemed for airport testing from the total number of travelers tested.§On August 29, 2020, airport programs switched from reporting data on a Saturday–Friday schedule to a Sunday–Saturday schedule, resulting in an 8-day report for that week.¶"Other" includes children aged <2 years (exempt from testing), critical infrastructure workers following an alternative workforce and community protection plan, and travelers who arrived with proof of 1) a positive test result within the past 90 days and 2) completion of isolation. Beginning August 11, 2020, children aged <10 years were also exempt from testing.

During June–September, <1.0% of airport test results were positive; this increased to 2.6% during October–November (Figure 2). Over the entire study period (June–November), 951 tests were positive (0.8% overall). The percentage of test results that were positive at airports was consistently lower than the overall percentage in Alaska.

Figure 2.

Percentage of positive SARS-CoV-2 test results among air travelers arriving from out of state, percentage of positive SARS-CoV-2 test results statewide, and weekly number of SARS-CoV-2 cases statewide, by specimen collection date* — 10 airports, Alaska, June 6–November 14, 2020*When specimen collection date was not available, the report date, date of hospitalization, or date of symptom onset was used, whichever was earliest.

In response to a November survey, airport testing program administrators reported that clear communication, preparation, and organization were important for operational success; challenges included managing travelers' expectations and ensuring sufficient personnel and physical space. For example, administrators reported that travelers were frequently unprepared for screening and that space limitations resulted in travelers being unable to maintain sufficient physical distance. One airport noted an improvement in passenger attitudes and their willingness to complete declaration forms after the initiation of a broad educational campaign for travelers, a hotline for travelers to ask questions, and targeted messaging for travelers before and during travel. Administrators also reported that the travel screening and testing program was resource-intensive. For example, during June–November, Alaska's largest airport had a weekly average of nearly 12,000 passengers and 51 out-of-state flight arrivals; this airport required up to 22 screening personnel and five testing personnel per day and performed an average of approximately 3,500 tests per week. The cost of this program was also substantial, with a budget of $26 million for June–December.