Chapter 19, Problem 11E

(a)

To determine

To construct a $95\%$ confidence interval.

Answer to Problem 11E

  $(0.798,0.863)$ .

Explanation of Solution

It is given in the question that, consumer reports published their study of bacterial contamination of chicken in the U.S. 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,

  $\begin{array}{l}np=435.75>10\\ n(1-p)=89.25>10\end{array}$

$10\%$ 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,

  $\begin{array}{l}x=436\\ n=525\\ c=0.95\end{array}$

Thus, the confidence interval will be as:

  $(0.798,0.863)$ .

(b)

To determine

To explain what your confidence interval says about chicken sold in the U.S.

Explanation of Solution

It is given in the question that, consumer reports published their study of bacterial contamination of chicken in the U.S. 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,

  $\begin{array}{l}np=435.75>10\\ n(1-p)=89.25>10\end{array}$

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

Thus, all the conditions are met. Therefore, the confidence interval is $(0.798,0.863)$ . Thus, we can conclude from it that we are $95\%$ confident that the interval $(0.798,0.863)$ contains the true population proportion of chicken in the U.S. that are contaminated with Campylobacter.

(c)

To determine

To explain is this criticism valid.

Answer to Problem 11E

No, it is not valid.

Explanation of Solution

It is given in the question that, consumer reports published their study of bacterial contamination of chicken in the U.S. Therefore, the confidence interval is $(0.798,0.863)$ . Thus, we can say that this criticism is not valid because our sample size is large enough to show a fairly accurate approximation.