A certain STAT 410 instructor claims that the average time to complete a STAT 41

Question

A certain STAT 410 instructor claims that the average time to complete a STAT 410 homework assignment is at most 150 minutes. Assume that the time to complete a STAT 410 homework assignment is approximately normally distributed with standard deviation sigma = 40 minutes. We wish to test H_0: mu lessthaorequalto 150 vs H_1: mu > 150. a) A random sample of n = 49 students yields the sample mean of 162 minutes. Find the p-value for the test. b) Find the Rejection Region for the test at a alpha = 0.05. That is, for which values of the sample mean X should we reject H_0, if a 5% level of significance is used? c) Find the power of the test if the actual value of the average time to complete a STAT 410 homework assignment is 153, 157, 161, and 165 minutes, and a 5% level of significance is used. d) Sketch the graph of the power function.

Explanation / Answer

a)

z = ( x - mean) / ( s /sqrt(n))

= ( 162 - 150) / ( 40 / sqrt(49))

= 2.1

p value is calculated using z = 2.1

p value = 0.017864.

b)

critical value for 0.05 = 1.64 for right tailed test if z > 1.64 then we rejecct H0

we reject Ho at 0.05 significance level

c)

If true mean is 153, power of test can be calculated as below

If mean of actual completion time is 157, power of test can be calculated as below

If actual mean = 161, power of test is calculated as below

If actual mean = 165, power of test is as below

mu0 (hypothesised mean) 150 sigma 40 n 49 alpha 0.05 sample/true mean 153 Std. Error. SE = sigma/sqrt(n) 5.7143 Zcritical -1.645 Xcritical 140.60 Beta or type II error is the probability of fail to reject the null hypothesis (X<140.6) 0.0150 Hence type II error probability is 0.015 Power of the test 0.9850

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