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Chapter 11: Problem 17

On April \(12,1955,\) Dr. Jonas Salk released the results of clinical trials for his vaccine to prevent polio. In these clinical trials, 400,000 children were randomly divided in two groups. The subjects in Group 1 (the experimental group) were given the vaccine, while the subjects in Group 2 (the control group) were given a placebo. Of the 200,000 children in the experimental group, 33 developed polio. Of the 200,000 children in the control group, 115 developed polio. (a) Test whether the proportion of subjects in the experimental group who contracted polio is less than the proportion of subjects in the control group who contracted polio at the \(\alpha=0.01\) level of significance. (b) Construct a \(90 \%\) confidence interval for the difference between the two population proportions, \(p_{1}-p_{2}\)

Short Answer

Expert verified
(a) Reject the null hypothesis, (b) 90% CI: \(-0.0004756 , -0.0003444\).

Step by step solution

01

Define null and alternative hypotheses for part (a)

We need to test if the proportion of children who contracted polio is less in the experimental group than in the control group. Let the proportion of children who contracted polio in the experimental group be denoted by \(p_1\) and in the control group by \(p_2\). The hypotheses are: \[\text{Null Hypothesis (H_0)}: p_1 \geq p_2\] \[\text{Alternative Hypothesis (H_a)}: p_1 < p_2\]
02

Calculate sample proportions

Calculate sample proportions for both groups. For the experimental group, \(\hat{p}_1 = \frac{33}{200,000} = 0.000165\). For the control group, \(\hat{p}_2 = \frac{115}{200,000} = 0.000575\).
03

Calculate the pooled sample proportion

Calculate the pooled sample proportion \(\hat{p}\) as follows: \[\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{33 + 115}{200,000 + 200,000} = \frac{148}{400,000} = 0.00037\]
04

Calculate the standard error

Calculate the standard error (SE) using the pooled proportion: \[SE = \sqrt{ \hat{p} \left(1 - \hat{p}\right) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{ 0.00037 \times (1-0.00037) \times \left( \frac{1}{200,000} + \frac{1}{200,000} \right) } = \sqrt{0.00037 \times 0.99963 \times 0.00001} = 0.0000606\]
05

Calculate the Z-test statistic

Calculate the Z-test statistic using the formula: \[Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.000165 - 0.000575}{0.0000606} = -6.77\]
06

Determine the critical value and make a decision

For a left-tailed test at \( \alpha = 0.01 \), the critical value (Z_{ \alpha }) is \(-2.33\). Since \(-6.77 < -2.33\), we reject the null hypothesis.
07

Interpret the result for part (a)

There is sufficient evidence at the \( \alpha = 0.01 \) level to conclude that the proportion of children vaccinated who contracted polio is less than the proportion of children in the control group who contracted polio.
08

Construct the 90% confidence interval for part (b)

To find the 90% confidence interval for the difference between population proportions \(p_1 - p_2\): \[MOE = Z_{ \alpha/2} \times SE = 1.645 \times \sqrt{\left( \frac{ \hat{p}_1 \left(1 - \hat{p}_1 \right)}{n_1} + \frac{ \hat{p}_2 \left(1 - \hat{p}_2 \right)}{n_2} \right)} = 1.645 \times \sqrt{\left( \frac{0.000165 \left(1 - 0.000165\right)}{200,000} + \frac{0.000575 \left(1 - 0.000575\right)}{200,000} \right)} = 1.645 \times 0.0000399 = 0.0000656\] Thus, the confidence interval is: \[ ( \hat{p}_1 - \hat{p}_2) \pm MOE = 0.000165 - 0.000575 \pm 0.0000656 = [ -0.0004756 , -0.0003444 ] \]
09

Interpret the confidence interval

We are 90% confident that the difference in the population proportions \(p_1 - p_2\) is between \(-0.0004756\) and \(-0.0003444\). This interval does not include 0, which supports the conclusion that the vaccine is effective in reducing the rate of polio.

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Key Concepts

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

null hypothesis
The null hypothesis, often denoted as \(\text{H_0}\), is a statement that assumes no effect or no difference exists in a population. In hypothesis testing, it acts as the default or starting assumption. For the given problem, the null hypothesis states that the proportion of children who contracted polio in the experimental group is greater than or equal to the proportion in the control group. Formally, it is written as \(\text{H_0}: p_1 \geq p_2\).

The main objective in hypothesis testing is to gather evidence to either reject or fail to reject the null hypothesis. If the test results show strong evidence against the null hypothesis, it can be rejected. However, if the evidence is not strong enough, the null hypothesis cannot be rejected, meaning it remains a plausible explanation.
alternative hypothesis
The alternative hypothesis, denoted as \(\text{H_a}\), is the statement that contradicts the null hypothesis. It represents the effect or difference that the researcher aims to prove. In this problem, the alternative hypothesis asserts that the proportion of children who contracted polio in the experimental group is less than the proportion in the control group. It is mathematically expressed as \(\text{H_a}: p_1 < p_2\).

The alternative hypothesis is what researchers usually aim to support through their experiments. If enough evidence is found against the null hypothesis, it supports the alternative hypothesis, suggesting there is a significant effect or difference.
confidence interval
A confidence interval is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. It provides an estimate of where the true population parameter lies and helps gauge the precision of the estimate. For instance, a 90% confidence interval suggests that if the same population were sampled multiple times, 90% of those intervals would contain the true population parameter.

In the problem, a 90% confidence interval for the difference between the two population proportions \(p_1 - p_2\) is constructed. This interval is calculated to range between \(-0.0004756\) and \(-0.0003444\). Since this range does not include 0, it indicates a significant difference between the two proportions, implying that the vaccine is effective in reducing the rate of polio.
proportion
A proportion represents a part or fraction of a whole. It is the ratio of a specific number of observations to the total number of observations. In hypothesis testing, proportions are often compared to determine if significant differences exist between groups.

In this exercise, the sample proportions are derived by dividing the number of children who contracted polio by the total number of children in each group. For the experimental group, the proportion is \(\frac{33}{200,000} = 0.000165\), and for the control group, it is \(\frac{115}{200,000} = 0.000575\). These proportions are then used in further calculations to test the hypotheses and construct confidence intervals.

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