Question

A sample of 300 urban adult residents of a particular state revealed 65 who favored increasing the highway speed limit from 55 to 65 mph, whereas a sample of 171 rural residents yielded 78 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 H0: p1 = p2 = 0 versus Ha: p1 = p2 = 0 using a = 0.05, where p1 refers to the urban population. (Round your test statistic to two decimal places and your P-value to four decimal places.)z = P-value = State the conclusion in the problem context.A. Reject H0. The data does not suggest that the sentiment for increasing the speed limit is different for the two groups of residents.B. Fail to reject H0. The data suggests that the sentiment for increasing the speed limit is different for the two groups of residents. C. Fail to reject H0. The data does not suggest that the sentiment for increasing the speed limit is different for the two groups of residents.D. Reject H0. The data suggests 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 p1 = 0.24 (urban) and p2 = 0.40 (rural), what is the probability that H0 will be rejected using a level 0.05 test with m = 297, n = 189? (Round your answer to four decimal places.)

Answer 1

Answer and explanation:

Null hypothesis(H0) says sentiment for increasing speed limit in the two populations is the same

Alternative hypothesis(Ha) says sentiment for increasing speed limit in the two populations is different

Where p1 is the first population proportion =65/297= 0.22

And p2 is the second population proportion = 78/189= 0.41

P= p1+p2/n1+n2= 65+78/297+189= 0.29

Hence H0= p1-p2=0

Ha=p1-p2≠0

Test statistic= p1-p2/√p(1-p) (1/n1+1/n2)

= 0.22-0.41/√0.29(1-0.29)(1/297+1/189)

= -4.49

Critical value at 95% significance level= 1.96( from tables)

We therefore reject null hypothesis as critical value is greater than test statistic

Therefore the sentiment for increasing speed limit in the two populations is different

b. at proportions of 0.24 and 0.40 for p1 and p2 respectively

Test statistic = 0.24-0.40/0.29(1-0.29)(1/297+1/189)

= -3.78

P value is 0.0002 at 0.05 significance level

Hence probability =0.4998