Chapter PV, Problem 39RE

To determine

To construct and interpret an appropriate confidence interval.

Answer to Problem 39RE

We are $95\%$ confident that it is between $6.95\%\text{ and 19}\text{.19\%}$ more Americans like the idea of number energy than like the idea of nuclear energy plants in their area.

Explanation of Solution

In this question, to find out how large is the difference between the proportion of American adults who think nuclear energy is good idea and the proportion who would be willing to have a nuclear plant in their area. Now, let us check the conditions and assumptions necessary for inference as:

Random condition: It is satisfied as the sample was randomly selected from the population.

Independent condition: It is satisfied as we can assume that the sample is independent.

  $10\%$ condition: It is satisfied as the adults are less than $10\%$ of all adults.

Success/failure condition: It is satisfied because both are greater than ten as,

  $\begin{array}{l}np=266>10\\ n(1-p)=236>10\end{array}$

  $\begin{array}{l}np=200>10\\ n(1-p)=301>10\end{array}$

Thus, all the conditions are satisfied. Therefore, now, to find the confidence interval we will use the calculator $\text{TI}-89$ . So, we have to first go to the STAT $\text{Ints}$ menu and select $6:2-\text{PropZInt}$ . Then we have to enter the observed counts and the sample sizes for both the samples. Then specify a confidence level. Then the interval will be calculated. Thus, we have,

  $\begin{array}{l}{x}_1=266\\ n_1=502\\ x_2=200\\ n_2=501\\ \alpha =0.05\end{array}$

Thus, the confidence interval will be as:

  $(0.06948,0.19188)$ . Thus, we are $95\%$ confident that it is between $6.95\%\text{ and 19}\text{.19\%}$ more Americans like the idea of number energy than like the idea of nuclear energy plants in their area.