It is thought that the front cover and the nature of the first question on mail

Question

It is thought that the front cover and the nature of the first question on mail surveys influence the response rate. An article tested this theory by experimenting with different cover designs. One cover was plain; the other used a picture of a skydiver. The researchers speculated that the return rate would be lower for the plain cover. Does this data support the researchers' hypothesis? Test the relevant hypotheses using alpha = 0.10 by first calculating a P-value. State the relevant hypotheses. (Use p_1 for the plain cover and p_2 for the skydiver cover.) H_0: p_1 - p_2 = 0 H_a: p_1 - p_2 0 Compute the test statistic value and find the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

Explanation / Answer

For sample 1, we have that the sample size is N1= 208, the number of favorable cases is X1=103, so then the sample proportion is p1 hat = X1/N1 = 103/208 = 0.4952

For sample 2, we have that the sample size is N2= 211, the number of favorable cases is X2=107, so then the sample proportion is p2 hat  = X2/N2 = 107/211 =0.5071

The value of the pooled proportion is computed as P hat = X1+X2/N1+N2 = 103+107/208+211 = 0.5012

Also, the given significance level is =0.10.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p1 - p2= 0

Ha: p1 - p2 <0

This corresponds to a left-tailed test, for which a z-test for two population proportions needs to be conducted

Rejection Region

Based on the information provided, the significance level is = 0.10, and the critical value for a left-tailed test is zc = 1.28.

Test Statistics

The z-statistic is computed as follows:

z = (p1 hat -  p2 hat) / sqrt (P hat (1- P hat)(1/N1+1/N2))

z = (0.4952 - 0.5071) / sqrt (0.5012 (1 - 0.5012)(1/208+1/211)

z = - 0.244

therefore z = -0.24

Decision about the null hypothesis

Since it is observed that z = -0.244 zc= 1.28, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.4036, and since p = 0.4036 0.10, it is concluded that the null hypothesis is not rejected.

Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p1 is less than p2, at the 0.10 significance level.

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