Chapter PV, Problem 24RE

(a)

To determine

To create a $90\%$ confidence interval for the level of cheating among high school students.

Answer to Problem 24RE

  $(0.72924,0.75076)$ .

Explanation of Solution

It is given in the question that a university study found that many high school students cheat on tests. Thus, the conditions and assumptions for inference is as follows:

Random condition: It is satisfied because the sample of students is randomly selected.

  $10\%$ condition: It is less than $10\%$ of all students.

Success/failure condition: It is satisfied as,

  $\begin{array}{l}np=3330>10\\ n(1-p)=1170>10\end{array}$

Independent condition: It is satisfied because students are independent of others.

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,

  $\begin{array}{l}x=3330\\ n=4500\\ c=0.90=90\%\end{array}$

Thus, the confidence interval will be as:

  $(0.72924,0.75076)$ .

(b)

To determine

To interpret your interval.

Explanation of Solution

It is given in the question that a university study found that many high school students cheat on tests. Thus, the confidence interval will be as:

  $(0.72924,0.75076)$ .

Thus, the confidence interval above means that we are $90\%$ confident that between $72.9\%\text{ and 75\%}$ of high school students have cheated on tests.

(c)

To determine

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

Explanation of Solution

It is given in the question that a university study found that many high school students cheat on tests. Thus, the confidence interval will be as:

  $(0.72924,0.75076)$ .

Thus, the confidence interval above means that we are $90\%$ confident that between $72.9\%\text{ and 75\%}$ of high school students have cheated on tests. Thus, $90\%$ confidence means in this context that $90\%$ of intervals created from samples will contain the true population proportions.

(d)

To determine

To explain a $95\%$ confidence interval be wider or narrower.

Answer to Problem 24RE

It will be wider.

Explanation of Solution

It is given in the question that a university study found that many high school students cheat on tests. Thus, the confidence interval will be as:

  $(0.72924,0.75076)$ .

Thus, the confidence interval above means that we are $90\%$ confident that between $72.9\%\text{ and 75\%}$ of high school students have cheated on tests. Thus, without calculating the confidence interval we can say that at $95\%$ confidence interval be wider because in order to be more sure we have to be less precise.