Question

In the initial test of the Salk vaccine for polio, 400,000 children were selected and divided into two groups of 200,000. One group was vaccinated with the Salk vaccine while the second group was vaccinated with a placebo. Of those vaccinated with the Salk vaccine, 33 later developed polio. Of those receiving the placebo, 115 later developed polio. Test the hypothesis that the Salk vaccine is effective in lowering the polio rate. Use a=0.01.

Construct a 99% confidence interval for the difference between the proportion of children who developed polio after taking the Salk vaccine and those who developed polio after taking the placebo. Interpret the result in a complete sentence.

Answer 1

Given :

x1 = 33

x2 = 115

n1 = 200,000

n2 = 200,000

Null and alternative hypothesis are

Test Statistic :

Now plug the values in the formula

Using Excel function = NORMSDIST(z)

=NORMSDIST( -6.74 )

=7.92E-12

which is approximately equal to zero .

So p-value = 0

Decision rule :

If p-value is less than the significance level () then we reject null hypothesis .

As p-value (0) is less than significance level (0.01 ) , we reject the null hypothesis .

Conclusion :

There is sufficient evidence to support the claim that the Salk vaccine is effective in lowering the polio rate.

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For 99% confidence interval

The formula for confidence interval of difference in proportion is

c = 0.99

Using Excel function

That is

Therefore 99% confidence interval for the difference between the proportion of children who developed polio after taking the Salk vaccine and those who developed polio after taking the placebo is ( - 0.00057 , -0.00025 )

Interpretation :

We are 99% confident that the difference between the proportion of children who developed polio after taking the Salk vaccine and those who developed polio after taking the placebo is is between - 0.00057 and -0.00025