9.45 In general, do we want the power corresponding to a serious Type II error t

Question

9.45

In general, do we want the power corresponding to a serious Type II error to be near 0 or near 1? Explain. Again consider the Consolidated Power waste water situation. Remember that the power plant will be shut down and corrective action will be taken on The cooling system if the null hypothesis H_0: mu 60 is rejected in favour of H_a : mu > 60. In this exercise we calculate probabilities of various Type II errors in the context of this situation. Recall that Consolidated Power's hypothesis test is based on a sample of n = 100 temperature readings and assume that sigma equals 2. If the power company sets alpha =.025. calculate the probability of a Type II error for each of the following alternative values of mu: 60.1. 60.2. 60.3.60.4. 60.5, 60.6. 60.7, 60.8, 60.9, 61. if we want the probability of making a Type II error when mu. equals 60.5 to be very small, is Consolidated Power's hypothesis test adequate? Explain why or why not. if not and if we wish to maintain the value of alpha at.025, what must be done? The power curve for a statistical test is a plot of the power = 1 - beta on the vertical axis versus values of mu that make the null hypothesis false on the horizontal axis. Plot the power curve for Consolidated Power's test of H_0: 60 versus H_sigma: mu > 60 by plotting power 1 - beta for cach of the alternative values of mu in part a. What happens to the power of the test as the alternative value of mu moves away from 60?

Explanation / Answer

a) ANSWER for µ: 60.1

Power and Sample Size

1-Sample t Test

Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.025 Assumed standard deviation = 2

Sample
Difference Size Power
0.1 100 0.0714884


ANSWER for µ: 60.2
Power and Sample Size
1-Sample t Test
Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.025 Assumed standard deviation = 2

Sample
Difference Size Power
0.2 100 0.166112


ANSWER for µ: 60.3
Power and Sample Size
1-Sample t Test
Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.025 Assumed standard deviation = 2

Sample
Difference Size Power
0.3 100 0.317570



ANSWER for µ: 60.4
Power and Sample Size
1-Sample t Test
Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.025 Assumed standard deviation = 2

Sample
Difference Size Power
0.4 100 0.508224



ANSWER for µ: 60.5
Power and Sample Size
1-Sample t Test
Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.025 Assumed standard deviation = 2

Sample
Difference Size Power
0.5 100 0.696976


ANSWER for µ: 60.6
Power and Sample Size
1-Sample t Test
Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.025 Assumed standard deviation = 2

Sample
Difference Size Power
0.6 100 0.843947



ANSWER for µ: 60.7
Power and Sample Size
1-Sample t Test
Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.025 Assumed standard deviation = 2

Sample
Difference Size Power
0.7 100 0.933952


ANSWER for µ: 60.8
Power and Sample Size
1-Sample t Test
Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.025 Assumed standard deviation = 2

Sample
Difference Size Power
0.8 100 0.977301


ANSWER for µ: 60.9
Power and Sample Size
1-Sample t Test
Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.025 Assumed standard deviation = 2

Sample
Difference Size Power
0.9 100 0.993720



b) ANSWER: For probability of Type II error when (mu) equals 60.5 to be very small, increase the Sample Size by a factor of about 5x.

For Sample Size = 500 which is 5x the original Sample Size (100); then:

Power and Sample Size

1-Sample t Test

Testing mean = null (versus > null)
Calculating power for mean = null + difference
Alpha = 0.025 Assumed standard deviation = 2


Sample
Difference Size Power
0.5 500 0.999852



c) ANSWER: As the alternative value of (mu) in part a. [above] move more distant from (mu) = 60, the power (= 1 - Beta) becomes larger. This means there is a larger likelihood of a Type II error.

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