Question

Suppose Jonathan works for the local transportation authority and knows that for the last few years, only 62% of buses have arrived at their destination on time. However, the city recently completed improvements to the main roads, and Jonathan thinks that a higher proportion of buses now arrive on time because of these improvements. Jonathan collects a random sample of bus arrival times for 500 buses and finds that 328 buses arrived on time. He conducts a one-sample, right-sided z-test for a proportion, using a significance level of a = 0.05. Jonathan wants to know the power of the test to reject the null hypothesis if the true proportion of on time arrivals is 65% or more. What is the minimum sample proportion ộ that rejects Ho at a = 0.05? Please give your answer as a decimal precise to at least four decimal places. f = Determine the power of the test to correctly reject the null hypothesis if the population proportion is actually 0.65. That is, compute the probability that if the population proportion is 0.65, the sample proportion would be at least as great as the value of ê you found in the first step. Please give your answer as a decimal precise to at least four decimal places. - power =

Answer 1

Answer:

a)

Given,

z = (p^ - p)/sqrt(pq/n)

substitute values

- 1.64 = (p^ - 0.62)/sqrt(0.62(1-0.62)/500)

p^ = 0.5844

b)

P(p > 0.5844) = P(z > (0.5844 - 0.65)/sqrt(0.65(1-0.65)/500))

= P(z > - 3.08)

= 0.998965 [since from z table]

= 0.9990