Chapter 22, Problem 15E

(a)

To determine

To find: the hypothesis test.

Answer to Problem 15E

  $\begin{array}{r}{\text{H}}_0:p_V-p_{UV}=0\\ {\text{H}}_{\text{A}}:p_V-p_{UV}<0\end{array}$

Explanation of Solution

This is where we want to test if the vaccine is effective. Suppose $p_V$ indicate the proportion of ear infections among vaccinated children, and $p_{UV}$ indicate the proportion of ear infections among unvaccinated children. If the vaccine is successful, $p_V$ should be lower than $p_{UV}$ and the $p_V$ - $p_{UV}$ difference should be lower than 0. So, hypotheses are going to be

  $\begin{array}{r}{\text{H}}_0:p_V-p_{UV}=0\\ {\text{H}}_{\text{A}}:p_V-p_{UV}<0\end{array}$

(b)

To determine

To State: a conclusion based on confidence interval.

Answer to Problem 15E

Reject the null hypothesis

Explanation of Solution

Since 0 is not in confidence interval and then both values are positive, it can infer that the proportion of ear infections among unvaccinated children is 4.7 to 8.88 percent higher than that of vaccinated children. it dismisses the null hypothesis, therefore, and conclude that the vaccine is effective.

(c)

To determine

To find: the test use of the alpha level.

Explanation of Solution

Since there have used a 95 percent confidence interval, one side is the alternative hypothesis. The alpha level test used is, thus,

  $\begin{array}{c}\alpha =\frac{100-95}{2}\%\\ =2.5\%\end{array}$

(d)

To determine

To find: the type of error makes if the inference is incorrect.

Answer to Problem 15E

Type I

Explanation of Solution

It is dismissed the null hypothesis here. If that inference is incorrect, it implies that although it is valid, the null hypothesis is rejected. So, we made an error with Type I.

(e)

To determine

To find: the consequences of a mistake of this kind.

Explanation of Solution

It is dismissed the null hypothesis here. If that inference is incorrect, it suggests that the vaccine was not effective and we gave a not effective vaccine to children.