Chapter 19, Problem 22E

(a)

To determine

To explain which one will have the larger margin of error.

Answer to Problem 22E

The class’s interval will be larger since its sample size is smaller.

Explanation of Solution

It is given in the question that a city ballot includes a local initiative that would legalize gambling. The issue is hotly contested and two groups decide to conduct polls to predict the outcome. Thus, we can say that the class’s interval will be larger since its sample size is smaller.

(b)

To determine

To find both confidence interval.

Answer to Problem 22E

For newspaper: $(0.50176,0.55824)$ .

For class: $(0.49395,0.58605)$ .

Explanation of Solution

It is given in the question that a city ballot includes a local initiative that would legalize gambling. The issue is hotly contested and two groups decide to conduct polls to predict the outcome. Thus, we can say that the class’s interval will be larger since its sample size is smaller. Thus, we have,

Let us first check the conditions for the newspaper as:

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

Normal condition: It is satisfied as,

  $\begin{array}{l}np=636>10\\ n(1-p)=564>10\end{array}$

$10\%$ 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 for newspaper 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,

  $\begin{array}{l}x=636\\ n=1200\\ c=0.95\end{array}$

Thus, the confidence interval for newspaper will be as:

  $(0.50176,0.55824)$

Now, for the class, first check the conditions as:

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

Normal condition: It is satisfied as,

  $\begin{array}{l}np=243>10\\ n(1-p)=207>10\end{array}$

$10\%$ 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 for class 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,

  $\begin{array}{l}x=243\\ n=450\\ c=0.95\end{array}$

Thus, the confidence interval for class will be as:

  $(0.49395,0.58605)$

(c)

To determine

To explain which group concludes that the outcome is too close to call and why.

Answer to Problem 22E

The students group concludes that the outcome is too close to call.

Explanation of Solution

It is given in the question that a city ballot includes a local initiative that would legalize gambling. The issue is hotly contested and two groups decide to conduct polls to predict the outcome. Thus, we can say that the class’s interval will be larger since its sample size is smaller. Thus, the confidence interval for newspaper will be as:

  $(0.50176,0.55824)$

And the confidence interval for class will be as:

  $(0.49395,0.58605)$

So, we can say that the group of students conclude that the outcome is too close to call because $50\%$ is included in their interval.