Chapter 19, Problem 24E

(a)

To determine

To find a $90\%$ confidence interval for the success rate at this clinic.

Answer to Problem 24E

  $(0.188,0.285)$ .

Explanation of Solution

It is given in the question that in San Diego reproductive clinic reported live births to women under age of $40$ who had previously been unable to conceive. Thus, we have,

Let us first check the conditions for the following as:

Random condition: It is satisfied as it is assumed that the sample is randomly selected.

Normal condition: It is satisfied as,

  $\begin{array}{l}np=49>10\\ n(1-p)=158>10\end{array}$

$10\%$ condition: It is satisfied as the sample size is less than $10\%$ of the population size.

Thus, all the conditions are met. Therefore, now, to find the confidence interval we will use the calculator $\text{TI}-89$ . So, we have to first go to the $\text{Ints}$ menu and select $5:1-\text{PropZInt}$ . Then we have to enter the number of successes observed and the sample size. Then specify a confidence level. Then the interval will be calculated. Thus, we have,

  $\begin{array}{l}x=49\\ n=207\\ c=0.90\end{array}$

Thus, the confidence interval will be as:

  $(0.188,0.285)$

(b)

To determine

To interpret your interval in this context.

Explanation of Solution

It is given in the question that in San Diego reproductive clinic reported live births to women under age of $40$ who had previously been unable to conceive. Thus, the confidence interval will be as:

  $(0.188,0.285)$

Thus, we conclude that we are $90\%$ confident that the interval $(0.188,0.285)$ contains the true population proportion of women under $40$ helped at this clinic that will become pregnant and give birth.

(c)

To determine

To explain what “ $90\%$ confidence” means.

Explanation of Solution

It is given in the question that in San Diego reproductive clinic reported live births to women under age of $40$ who had previously been unable to conceive. Thus, the confidence interval will be as:

  $(0.188,0.285)$

Thus, we conclude that we are $90\%$ confident that the interval $(0.188,0.285)$ contains the true population proportion of women under $40$ helped at this clinic that will become pregnant and give birth. Thus, “ $90\%$ confidence” means that about $90\%$ of all random samples will produce a confidence interval that contains the true proportion.

(d)

To determine

To explain do these data refute the clinic’s claim of a $25\%$ success rate.

Answer to Problem 24E

These data do not refute the clinic’s claim of a $25\%$ success rate.

Explanation of Solution

It is given in the question that in San Diego reproductive clinic reported live births to women under age of $40$ who had previously been unable to conceive. Thus, the confidence interval will be as:

  $(0.188,0.285)$

Thus, we conclude that we are $90\%$ confident that the interval $(0.188,0.285)$ contains the true population proportion of women under $40$ helped at this clinic that will become pregnant and give birth. Thus, we can say that these data do not refute the clinic’s claim of a $25\%$ success rate because $0.25$ is contained in this interval.