Chapter 19, Problem 21E

(a)

To determine

To find out what kind of bias may be present here.

Answer to Problem 21E

It is a wording bias.

Explanation of Solution

It is given in the question that in the survey on the death penalty, the Gallup poll actually split the sample at random and asking the respondents to answer the question quoted earlier. Thus, according to the information provided by the question we can conclude that the bias that may be present in this is wording bias as they are asking the questions from the respondents.

(b)

To determine

To find out what is a $95\%$ confidence interval for the proportion of the general public that thinks the death penalty is being fairly applied.

Answer to Problem 21E

  $(54\%,60\%)$ .

Explanation of Solution

It is given in the question that in the survey on the death penalty, the Gallup poll actually split the sample at random and asking the respondents to answer the question quoted earlier. 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,

  $\begin{array}{l}np=510>10\\ n(1-p)=510>10\end{array}$

$10\%$ 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,

  $\begin{array}{l}x=510\\ n=1020\\ c=0.95\end{array}$

Thus, the confidence interval will be as:

  $(54\%,60\%)$

(c)

To determine

To explain how does the margin of error based on this pooled sample compare with the margins of error from the separate groups and why.

Answer to Problem 21E

It is smaller.

Explanation of Solution

It is given in the question that in the survey on the death penalty, the Gallup poll actually split the sample at random and asking the respondents to answer the question quoted earlier. Thus, the confidence interval will be as:

  $(54\%,60\%)$

Thus, we can say that the margin of error based on this pooled sample compare with the margins of error from the separate groups is smaller and a large sample size is there.