A sample of 289 urban adult residents of a particular state revealed 58 who favo

Question

A sample of 289 urban adult residents of a particular state revealed 58 who favored increasing the highway speed limit from 55 to 65 mph, whereas a sample of 176 rural residents yielded 70 who favored the increase. Does this data indicate that the sentiment for increasing the speed limit is different for the two groups of residents? (a) Test H_0: p_1 - p_2 = 0 versus H_a: p_1 - p_2 notequalto 0 using alpha = 0.05, where p_1 refers to the urban population. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = -4.61304 P-value =0 State the conclusion in the problem context. Fail to reject H_0. The data suggests that the sentiment for increasing the speed limit is different for the two groups of residents. Reject H_0. The data suggests that the sentiment for increasing the speed limit is different for the two groups of residents. Fail to reject H_0. The data does not suggest that the sentiment for increasing the speed limit is different for the two groups of residents. Reject H_0. The data does not suggest that the sentiment for increasing the speed limit is different for the two groups of residents. (b) If the true proportions favoring the increase are actually p_1 = 0.23 (urban) and p_2 = 0.42 (rural), what is the probability that H_0 will be rejected using a level 0.05 test with m = 289, n = 176? (Round your answer to four decimal places.) 0.

Explanation / Answer

(a) p1 = 58/289 = 0.20 and p2 = 70/176 = 0.398

Here Hypothesis are

H0 : p1-p2 = 0

H1 : p1 - p2 0

p* = (58+70)/(289 + 176) = 0.2753

Z = (0.398 - 0.200)/sqrt ( 0.2752 * 0.7247 * (1/289 + 1/176) = 0.198 / 0.0427 = - 4.636

P value = 0

option B is correct.

(b) Here true proportions are p1 = 0.23 and p2 = 0.42

p* = (289 * 0.23 + 176 * 0.42)/ ( 289 + 176) = 0.3019

so Z = (0.42 - 0.23)/ sqrt [ (0.302 * 0.698 * ( 1/289 + 1/176) ]

= 0.19/ 0.0439 = - 4.328

The probability that H0 will be rejected P( null hypothesis will be rejected ) = 1 - P(null hypothesis accepted)

P - value here = 1.5E-05

so probability that H0 will be rejected = 1- 1.5E-05 = 0.9999 or say 1

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