The actual arrival time of the commuter bus at the final bus stop as compared to

Question

The actual arrival time of the commuter bus at the final bus stop as compared to the scheduled arrival time is known to be Normally distributed with a mean of 1 minute and a standard deviation of 3 minutes (a negative value meaning the bus arrived early, a positive value meaning the bus arrived late). You ride the bus frequently and are under the impression that the mean arrival time is much later than they claim. You decide to test the hypotheses H0: = 1 versus Ha: > 1. You take a simple random sample of 10 trips and record the differences between the actual arrival time and the scheduled arrival time. You decide to reject H0 if > 3. If, in fact, the true mean is 2 minutes, what is the probability of a Type II error?

Explanation / Answer

here std error of mean =std deviaiton/(n)1/2 =3/(10)1/2 =0.9487

for probability of type II error is fail to rejeect null hypothesis

hence probability of a Type II error =P(Xbar<3 )=P(Z<(3-2)/0.9487)=P(Z<1.0541)=0.8541

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