Critical Thinking: Testing the Salk Vaccine The largest health experiment ever conducted involved a test of the Salk vaccine designed to protect children from the devastating effects of polio. The test included 201,229 children who were given the Salk vaccine, and 33 of them developed polio. The claim that the Salk vaccine is effective is equivalent to the claim that the proportion of vaccinated children who develop polio is less than 0.0000573, which was the rate of polio among children not given the Salk vaccine. (Note: The actual Salk vaccine experiment involved another group of 200,745 children who were injected with an ineffective salt solution instead of the Salk vaccine. This study design with a treatment group and placebo group is very common and very effective. Methods for comparing two proportions are presented in Chapter 9.)

Analyzing the Results

a. Test the given claim using a 0.05 significance level. Does the Salk vaccine appear to be effective?

b. For the hypothesis test from part (a), consider the following two errors:

• Concluding that the Salk vaccine is effective when it is not effective.

• Concluding that the Salk vaccine is not effective when it is effective. Determine which of the above two errors is a type I error and determine which is a type II error. Which error would have worse consequences? How could the hypothesis test be conducted in order to reduce the chance of making the more serious error?

In a sample of 201229 children who were vaccinated, 33 developed polio.

It is claimed that less than 0.0000573 vaccinated children developed polio.

The null hypothesis is written as follows:

The proportion of vaccinated children who developed polio is equal to 0.0000573.

\({H_0}:p = 0.0000573\)

The alternative hypothesis is written as follows:

The proportion of vaccinated children who developed polio is less than 0.0000573.

\({H_1}:p < 0.0000573\)

The test is left-tailed.

The sample size (n) is equal to 201229.

The sample proportion of vaccinated children who developed polio is equal to:

\(\begin{array}{c}\hat p = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{children}}\;{\rm{who}}\;{\rm{developed}}\;{\rm{polio}}}}{{{\rm{Sample}}\;{\rm{size}}}}\\ = \frac{{33}}{{201229}}\\ = 0.0001640\end{array}\)

The population proportion of vaccinated children who developed polio is equal to 0.0000573.

The value of the test statistic is computed below:

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.0001640 - 0.0000573}}{{\sqrt {\frac{{0.0000573\left( {1 - 0.0000573} \right)}}{{201229}}} }}\\ = 6.323\end{array}\)

Thus, z=6.323.

Referring to the standard normal table, the critical value of z at\(\alpha = 0.05\)for a left-tailed test is equal to -1.645.

Referring to the standard normal table, the p-value for the test statistic value of 6.323 is equal to 0.9999.

Since the p-value is greater than 0.05, so the decision is it fails to reject the null hypothesis.

a.

There is not enough evidence to support the claim that the proportion of vaccinated children who develop polio is less than 0.0000573.

Thus, it is concluded that the Salk vaccine is not effective.

b.

A type I error occurs by rejecting the null hypothesis when it is actually true.

A type II error occurs by failing to reject the null hypothesis when it is actually false.

Here, the hypotheses are as follows:

Null Hypothesis: The Salk vaccine is not effective.

Alternative Hypothesis: The Salk vaccine is effective.

Here, concluding that the Salk vaccine is effective when it is not effective will result in a Type I error. Moreover, concluding that the Salk vaccine is not effective when it is effective will result in a Type II error.

A vaccine is directly related to the health of children. It is created to lower the chances of disease in the target population. Accepting a poor vaccine as effective can lead to grave results.

Therefore, the type I error can have more serious consequences compared to the type II error.

The type I error is denoted by\(\alpha \)which is also the level of significance of the hypothesis test.

Thus, to reduce the chance of the more serious error (type I error), the value of the level of significance \(\left( \alpha \right)\) chosen should be as low as possible.