Chapter 20, Problem 15E

(a)

To determine

To find out what are the hypotheses.

Answer to Problem 15E

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p<0.05\end{array}$

Explanation of Solution

It is given in the question that an organization send free mailing labels and greeting cards to potential donors on their list and ask for a voluntary contribution. They have had a contribution rate of $5\%$ in past campaigns, but the staff worries that the rate will be lower. Thus, the null hypothesis will be that the rate will be equal to $5\%$ . And the alternative hypothesis is that the rate will be lower than $5\%$ . So, we define hypotheses as:

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p<0.05\end{array}$

(b)

To determine

To explain are the assumptions and conditions for inference met.

Answer to Problem 15E

Yes, all the assumptions and conditions for inference met.

Explanation of Solution

It is given in the question that an organization send free mailing labels and greeting cards to potential donors on their list and ask for a voluntary contribution. They have had a contribution rate of $5\%$ in past campaigns, but the staff worries that the rate will be lower. So, we define hypotheses as:

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p<0.05\end{array}$

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 sample is less than $10\%$ of all population.

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

  $\begin{array}{l}np=0.05(100000)>10\\ n(1-p)=0.95(100000)>10\end{array}$

Thus, all the conditions are satisfied.

(c)

To determine

To explain do you think the rate would drop.

Answer to Problem 15E

Yes, the rates will drop because the hypothesis test provides significant statistical evidence that rates will be lower than $0.05$ .

Explanation of Solution

It is given in the question that an organization send free mailing labels and greeting cards to potential donors on their list and ask for a voluntary contribution. They have had a contribution rate of $5\%$ in past campaigns, but the staff worries that the rate will be lower. So, we define hypotheses as:

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p<0.05\end{array}$

Thus, for this first we have to select $5:1-\text{PropZInt}$ from the STAT TESTS menu. Then, enter the observed count and sample size for the sample. Then we will indicate what kind of test we want: one-tail lower tail, two-tail, or one-tail upper tail. Then, we will specify whether to calculate the result or draw the result. Hence we have,

  $\begin{array}{l}x=5000\\ n=100000\\ c=0.95=95\%\end{array}$

So, by putting the values above by using the calculator we will get the result as,

The value of z-statistics and the P-value will be then,

  $\begin{array}{l}z=-3.178\\ P\approx 0\end{array}$

And as we know if the P-value is smaller than the significance level, then reject the null hypothesis, so, we have,

  $P<0.05\Rightarrow \text{ Reject }{H}_0$

Thus, we conclude that there is sufficient evidence that the rates will drop because the hypothesis test provides significant statistical evidence that rates will be lower than $0.05$ .