Suppose Jonathan works for the local transportation authority and knows that for

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 347 buses arrived on time. He conducts a one-sample z test for a proportion to test the null hypothesis, H 0 :p=0.62 , against the alternative hypothesis, H 1 :p>0.62 . Using a significance level of =0.01 , 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 p that rejects H 0 at = 0.01? Please give your answer as a decimal precise to at least four decimal places.

Explanation / Answer

Solution:

From the given data

p0 = 347/500 = 0.694

S.d. = sqrt(0.62*0.38/500) = 0.0217

Let min proportion be p that rejects H0 at alpha = 0.01

Zstat = (p - 0.62)/0.0217

Z0.01 = 2.57

To reject null hypothesis, Zstat > Z0.01

So, p - 0.62 > 2.57*0.0217

P > 0.0558 + 0.62

p > 0.6758

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