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Chapter 9: Problem 1

In the largest clinical trial ever conducted, 401,974 children were randomly assigned to two groups. The treatment group consisted of 201,229 children given the Salk vaccine for polio, and 33 of those children developed polio. The other 200,745 children were given a placebo, and 115 of those children developed polio. If we want to use the methods of this section to test the claim that the rate of polio is less for children given the Salk vaccine, are the requirements for a hypothesis test satisfied? Explain.

Short Answer

Expert verified
Yes, the requirements for a hypothesis test are satisfied because the sample sizes and the numbers of successes and failures meet the necessary conditions.

Step by step solution

01

- Identify the claim

The claim is that the rate of polio is less for children given the Salk vaccine. In other words, the proportion of children who develop polio in the vaccinated group is lower than in the placebo group.
02

- State the null and alternative hypotheses

Translate the claim into statistical hypotheses. The null hypothesis (H0) is that there is no difference in the polio rates between the two groups: H0: p1 = p2 The alternative hypothesis (H1) is that the polio rate is lower in the vaccinated group: H1: p1 < p2 where p1 is the proportion of children who develop polio in the vaccinated group, and p2 is the proportion in the placebo group.
03

- Check the sample size requirement

Ensure the sample sizes are sufficient for the test. In this case, both group sizes are large (201,229 in the treatment group and 200,745 in the placebo group), satisfying the requirement that they should each be at least 30.
04

- Check the number of successes and failures

Verify the number of observed successes (children who developed polio in this context) and failures in both groups. For the vaccine group, the number of successes is 33, and the number of failures is 201,229 - 33 = 201,196. Both numbers are greater than 5.For the placebo group, the number of successes is 115, and the number of failures is 200,745 - 115 = 200,630. Both numbers are also greater than 5. This satisfies the requirement.
05

- Conclusion on requirements

Since both the sample size and observed successes and failures meet the necessary conditions, the requirements for conducting a hypothesis test are satisfied.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Clinical Trials
Clinical trials are research studies performed in people aimed at evaluating a medical, surgical, or behavioral intervention. One of their main goals is to determine if new treatments are safe and effective.
In a large-scale clinical trial, such as the one involving the Salk vaccine for polio, it鈥檚 crucial to randomly assign participants to treatment and control groups to eliminate bias. This ensures the results are due to the intervention and not some other variable.
In our case study, 401,974 children were divided into two groups: one received the Salk vaccine, and the other received a placebo. Randomization ensures both groups are comparable, which is essential for validating the outcomes of the trial.
Null and Alternative Hypotheses
Hypothesis testing begins with forming the null and alternative hypotheses. The null hypothesis (H0) often states that there is no effect or no difference. In our context: H0: p1 = p2, where p1 and p2 represent the polio rates in the vaccinated and placebo groups, respectively.
The alternative hypothesis (H1) contradicts H0 and indicates what we aim to prove. Here, H1: p1 < p2, suggesting that the Salk vaccine reduces polio rates compared to the placebo.
By comparing these two hypotheses, researchers can determine if any observed difference in polio rates is statistically significant or if it might have happened by chance.
Sample Size Requirements
Sample size is critical in hypothesis testing because larger samples provide more reliable results. A rule of thumb is to have at least 30 participants in each group. In our study, the Salk vaccine trial had 201,229 children in the treatment group and 200,745 in the placebo group.
These large sample sizes ensure the statistical power of the study, increasing the accuracy of the hypothesis test results.
Additionally, large sample sizes help to minimize the margin of error and provide more confidence that the results reflect the true effect of the intervention.
Successes and Failures in Statistics
In statistics, 'successes' and 'failures' are terms used to describe the outcomes of binary trials. Success does not always imply a positive outcome; it merely represents one of two possible results. For the Salk vaccine trial, a 'success' means a child developed polio.
To perform a hypothesis test reliably, both the number of successes and failures in each group should be greater than 5. In the treatment group, we had 33 successes and 201,196 failures. In the placebo group, there were 115 successes and 200,630 failures.
Since all these numbers exceed 5, the sample data meet the necessary requirements, allowing us to conduct a valid hypothesis test.
  • This step ensures the results are not anomalies and reflect the actual behavior of the population studied.

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Most popular questions from this chapter

In one segment of the TV series MythBusters, an experiment was conducted to test the common belief that people are more likely to yawn when they see others yawning. In one group, 34 subjects were exposed to yawning, and 10 of them yawned. In another group, 16 subjects were not exposed to yawning, and 4 of them yawned. We want to test the belief that people are more likely to yawn when they are exposed to yawning. a. Why can't we test the claim using the methods of this section? b. If we ignore the requirements and use the methods of this section, what is the \(P\) -value? How does it compare to the \(P\) -value of \(0.5128\) that would be obtained by using Fisher's exact test? c. Comment on the conclusion of the Mythbusters segment that yawning is contagious.

Listed below are the numbers of words spoken in a day by each member of six different couples. The data are randomly selected from the first two columns in Data Set 24 "Word Counts" in Appendix B. a. Use a \(0.05\) significance level to test the claim that among couples, males speak fewer words in a day than females. b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)? $$ \begin{array}{l|l|l|r|r|r|r|r|r} \hline \text { Male } & 15,684 & 26,429 & 1,411 & 7,771 & 18,876 & 15,477 & 14,069 & 25,835 \\ \hline \text { Female } & 24,625 & 13,397 & 18,338 & 17,791 & 12,964 & 16,937 & 16,255 & 18,667 \\ \hline \end{array} $$

As part of the National Health and Nutrition Examination Survey, the Department of Health and Human Services obtained self-reported heights (in.) and measured heights (in.) for males aged 12-16. Listed below are sample results. Construct a \(99 \%\) confidence interval estimate of the mean difference between reported heights and measured heights. Interpret the resulting confidence interval, and comment on the implications of whether the confidence interval limits contain \(0 .\) $$ \begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { Reported } & 68 & 71 & 63 & 70 & 71 & 60 & 65 & 64 & 54 & 63 & 66 & 72 \\ \hline \text { Measured } & 67.9 & 69.9 & 64.9 & 68.3 & 70.3 & 60.6 & 64.5 & 67.0 & 55.6 & 74.2 & 65.0 & 70.8 \\ \hline \end{array} $$

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, \(P\) -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. A trial was conducted with 75 women in China given a 100 -yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from "The Denomination Effect," by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a \(0.05\) significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. If the significance level is changed to \(0.01\), does the conclusion change?

Data Set 26 "Cola Weights and Volumes" in Appendix B includes weights (lb) of the contents of cans of Diet Coke \((n=36, \bar{x}=0.78479 \mathrm{lb}, s=0.00439 \mathrm{lb})\) and of the contents of cans of regular Coke \((n=36, \bar{x}=0.81682 \mathrm{lb}, s=0.00751 \mathrm{lb})\). a. Use a \(0.05\) significance level to test the claim that the contents of cans of Diet Coke have weights with a mean that is less than the mean for regular Coke. b. Construct the confidence interval appropriate for the hypothesis test in part (a). c. Can you explain why cans of Diet Coke would weigh less than cans of regular Coke?
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