Chapter 19, Problem 16E

(a)

To determine

To give a $95\%$ confidence interval for the true proportion of those cardholders who will register for the offer.

Answer to Problem 16E

We conclude that we are $95\%$ confident that the interval $(0.02235,0.02501)$ contains the true population proportion of people who will register.

Explanation of Solution

It is given in the question that First USA, a major credit card company is planning a new offer for their current cardholders. The offer will give double airline miles on purchase for the next six months if the cardholder goes online and registers for the offer. 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,

  $\begin{array}{l}np=1184>10\\ n(1-p)=48816>10\end{array}$

$10\%$ 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,

  $\begin{array}{l}x=1184\\ n=50000\\ c=0.95\end{array}$

Thus, the confidence interval will be as:

  $(0.02235,0.02501)$

Thus, we conclude that we are $95\%$ confident that the interval $(0.02235,0.02501)$ contains the true population proportion of people who will register.

(b)

To determine

To explain what would you say, given the confidence interval.

Explanation of Solution

It is given in the question that First USA, a major credit card company is planning a new offer for their current cardholders. The offer will give double airline miles on purchase for the next six months if the cardholder goes online and registers for the offer. Thus, the confidence interval will be as:

  $(0.02235,0.02501)$

Thus, we conclude that we are $95\%$ confident that the interval $(0.02235,0.02501)$ contains the true population proportion of people who will register. So, given the confidence interval we would say that it is worth the expense because the expected amount of people who will register is greater than $0.02$ as shown by the confidence interval.