Question

1. A politician wants to open a new park in the middle of town. She needs to make sure that the local community supports the plan. A nearby town recently found that 60% of the community supported a similar plan and she believes that the same proportion will support it in her district. She takes a random survey of 40 people and find that the 10 people are supportive of her plan. She wants to be certain that this is not just due to statistical error. She is going to use an alpha of .01 for her study.
   1. What is the null and alternative hypotheses? (2)
   2. What are the critical values for this study? Please list all relevant information. (2)
   3. What is the statistic for the sample? (1)
   4. What is the P value for this sample (2)
   5. Do we reject the null hypotheses? Please interpret the hypotheses. (3)

Answer 1

The following information is provided: The sample size is N = 40 , the number of favorable cases is X = 10 , and the sample proportion is , and the significance level is α=0.01

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p = 0.6

Ha: p ≠​0.6

This corresponds to a two-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=0.01, and the critical value for a two-tailed test is z_c = 2.58

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that |z| = 4.518 >zc​=2.58, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0 , and since p = 0 <0.01, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p is different than p0​, at the α=0.01 significance level.

Hence, this is not just due to statistical errors'