Calculating the probability of a Type II error under a specified alternative A l

Question

Calculating the probability of a Type II error under a specified alternative A large insurance company has a complaint department where customers can call and speak to a representative. The company claims that the average time they keep their customers "on hold" before they reach a human is only 14 minutes with a SD= 6 minutes. I believe the true average hold time is much higher, at least 16.6 minutes, so I choose HA: True Average 16.6 minutes and assume the SD is still 6 minutes. To test their claim, I plan to call the complaint department at 25 randomly selected times and record the average time I’m kept on hold.

a. I decide to set the significance level of = 0.05 (5% in the tail). That means the probability of making a Type (1 or 2) error is _______ %. (Be sure to answer as a %, not a decimal.)

This corresponds to a Z cut-off= __________ Hint: This is just finding the Z score for a middle area=90%. for a  (1 or 2) tailed test.

c. Look at how the null cut-off divides ZD into 2 parts: ZD = |Z| + |Z|. Solve for |Z|.

|Z|= ________ Hint: You know |Z|and ZD so just solve for |Z|

d. How likely would it be for our test to fail to reject the null when the alternative is true (that the true average is really 16.6 minutes)? = ________ % (Answer as % between 0 and 50. Round to 1 decimal place. Use the p-value calculator on our course website to find the left hand tail (you can try using the normal table at the back of the notebook but it's safer to use the calculator because of rounding issues). But in using the calculator, be careful to include the correct sign of Z, not |Z|.)

e. What is the power of the test to detect the difference between H0 and HA? Power= __________%

Explanation / Answer

a) Probability of making a type 1 error is alpha = 0.05 or 5%.

b) The corresponding value of z-alpha = 1.64(from std normal dist table) for alpha = 0.05

c) The value of ZD = (X_bar-mu)/(sigma/sqrt(n)) = (16.6-14)/(6/5) = 2.166

now, z-beta = ZD - z-alpha = 2.166 - 1.64 = 0.526

d) the value of beta would be the p-value corresponding to z-beta, so beta = 0.3015(from std normal dist table) or 30.1%

e) the power of the test is 1- beta i.e. 1-0.3015 = 0.6985 or 69.85%

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