Questions: A direct mail appeal for contributions from a university's alumni and supporters is considered to be too costly if less than 27% of the alumni and supporters provide monetary contributions. To determine if a direct mail appeal is cost effective, the fundraising director sends the direct mail brochures to a simple random sample of 349 people on the alumni and supporters mailing lists. They receive monetary contributions from 80 people. Does this evidence demonstrate that the direct mail campaign is not cost effective? Use a 0.02 level of significance. Step 1 of 3 : State the null and alternative hypotheses for the test. Fill in the blank below. H0: p=0.27 Ha: p < 0.27

A direct mail appeal for contributions from a university's alumni and supporters is considered to be too costly if less than 27% of the alumni and supporters provide monetary contributions. To determine if a direct mail appeal is cost effective, the fundraising director sends the direct mail brochures to a simple random sample of 349 people on the alumni and supporters mailing lists. They receive monetary contributions from 80 people. Does this evidence demonstrate that the direct mail campaign is not cost effective? Use a 0.02 level of significance.

Step 1 of 3 : State the null and alternative hypotheses for the test. Fill in the blank below.

H0: p=0.27 Ha: p < 0.27

Transcript text: A direct mail appeal for contributions from a university's alumni and supporters is considered to be too costly if less than $27 \%$ of the alumni and supporters provide monetary contributions. To determine if a direct mail appeal is cost effective, the fundraising director sends the direct mail brochures to a simple random sample of 349 people on the alumni and supporters mailing lists. They receive monetary contributions from 80 people. Does this evidence demonstrate that the direct mail campaign is not cost effective? Use a 0.02 level of significance. Step 1 of 3 : State the null and alternative hypotheses for the test. Fill in the blank below. \[ \begin{array}{l} H_{0}: p=0.27 \\ H_{a}: p \quad 0.27 \end{array} \]

Solution

Solution Steps

Step 1: State the null and alternative hypotheses

Null hypothesis ($H_0$): $p = p_0$, where $p$ is the true proportion of alumni and supporters who provide contributions, and $p_0$ is the threshold proportion.
Alternative hypothesis ($H_a$): $p < p_0$, since the campaign is considered not cost-effective if the true proportion is less than the threshold.

Step 2: Compute the test statistic

The sample proportion ($\hat{p}$) is calculated as $\hat{p} = \frac{80}{349} = 0.229$.
The test statistic ($Z$) is calculated using the formula $Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} = -1.716$.

Step 3: Determine the critical value or p-value

The p-value corresponding to the test statistic from the standard normal distribution is 0.0431.

Step 4: Make a decision

Since the p-value (0.0431) is greater than or equal to the level of significance ($\alpha = 0.02$), we do not reject the null hypothesis. This indicates insufficient evidence to conclude that the campaign is not cost-effective.