Question

The Arizona Department of Transportation (ADoT) is considering changing statewide speed limits from 65 mph to 75 mph. They wish to perform analysis on ?, the proportion of Arizona residents that favor this change. In order to prove that a majority of residents favor the change, ADoT wishes to test the following hypotheses using a level of significance of 0.05.

H0: ? = 0.5

H1:? > 0.5

ADoT wishes to design the test to have a power of 90% if the true population proportion is equal to 0.55.

Part a : How many residents should be surveyed? You may use this sample size for the remainder of this problem.

Part b : Suppose the sample size found in part a is implemented and 461 of the surveyed residents favor the change. Construct a point estimate on the proportion of interest.

Part c : Identify the test statistic and critical region for the test. State the conclusion for this test.

Part d : What is the P-value for this test? State the conclusion for this test.

Part e : Construct an appropriate confidence interval to verify your conclusion from part c.

Part f : We wish to use a 95% lower confidence interval to estimate p. How many residents should be surveyed so that the error in our estimate is less than 4%.

Answer 1

Sol:

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p = 0.50 Ha:p* 0.50 This corresponds to a two-tailed test, for which a z-test for one population proportion needs to be used =1 (2) Rejection Region Based on the information provided, the significance level is a = 0.1, and the critical value for a two- tailed test ise = 1.64. The rejection region for this two-tailed test is R = {* : -|>1.64} (3) Test Statistics The 2-statistic is computed as follows: P-Po 0.55 -0.50 Po (1 - Pol/n 0.50(1 – 0.50)/100 (4) Decision about the null hypothesis Since it is observed that :=15= 1.64, it is then concluded that the null hypothesis is not rejected. Using the P-value approach: The p-value is p = 0.3173, and since p = 0.3173 > 0.1, 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 different than po, at the a = 0.1 significance level. Confidence interval The 90% confidence interval for p is: 0.468 <p <0.632.

$normaldistributiongrapher.php?mean=0&sig$

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