Question

Marketing again wants to know the proportion of people willing to pay for shipping on “free” products with their advertisement. They took a sample of 525 individuals and found that 345 of them were willing to pay for shipping on a “free” product. Using this information, marketing has asked if the true proportion of people who appeal to the ad is greater than 78%.

Z Test of Hypothesis for the Proportion
Data
Null Hypothesis            p =	0.78
Level of Significance	0.1
Number of Items of Interest	345
Sample Size	525
Intermediate Calculations
Sample Proportion	0.657142857
Standard Error	0.0181
Z Test Statistic	-6.7955
Upper-Tail Test
Upper Critical Value	1.2816
p-Value	1.0000
Do not reject the null hypothesis

Answer 1

The following information is provided: The sample size is N=525, the number of favorable cases is X=345, and the sample proportion is p = {345}/{525} = 0.6571, and the significance level is α=0.10

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p = 0.78

Ha: p > 0.78

This corresponds to a right-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.10, and the critical value for a right-tailed test is z_c​=1.28.

Here z value is calculated using standard normal distribution table

The rejection region for this right-tailed test is R={z:z>1.28}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z=−6.796 ≤ zc​=1.28, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=1, and since p=1≥0.10, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p is greater than p0​, at the α=0.10 significance level.