Question

In Exercise 33, you used a confidence interval to examine the effectiveness of a vaccine against ear infections in babies. Suppose that instead you had conducted a hypothesis test. (Answer these questions withoutactually doing the test.)
a) What hypotheses would you test?
b) State a conclusion based on your confidence interval.
c) If that conclusion is wrong, which type of error did you make?
d) What would be the consequences of such an error?

Answer 1

Answer:

Part a: The two hypotheses are $H_o: p_1=p_2$ and $H_1: p_1 < p_2$.

Part b: The Null hypothesis is rejected and Vaccines appear effective

Part c: As in this case the null hypothesis is rejected in the part b, thus the only possibility of error is type I error.

Part d:Children will get a vaccine which is not effective.

Explanation:

Part a:

The two hypothesis will be tested as indicated as  null hypothesis and alternative hypothesis

The null Hypothesis is given as that the population proportion for both the groups is same such that

$H_o: p_1=p_2$

The alternative hypothesis is the percentage of people who have got vaccination is less than those who have not got the vaccination such that

$H_1: p_1 < p_2$

So the two hypotheses are $H_o: p_1=p_2$ and $H_1: p_1 < p_2$.

Part b:

The confidence interval in the exercise 33 is calculated as (-0.008,-0.0470) as it does not contain the value of zero thus the difference is present which indicates that the two proportions are not same

Thus the Null hypothesis is rejected and Vaccines appear effective.

Part c:

There are two types of errors as

Type I where the null hypothesis H_o is wrongly rejected and it is true.

Type II where the null hypothesis H_o is not rejected and it is false.

As in this case the null hypothesis is rejected in the part b, thus the only possibility of error is type I error.

Part d:

If the conclusion is wrong this indicates that the null hypothesis was true. i.e. There is no difference in population percentage of children with or without vaccination. This means that children will get a vaccine which is not effective.