Explanation Given: α = 0.05 , with 0.30 population proportion. Explanation: The sample proportion is distributed normally with average μ p ^ = 0.30 and standard deviation σ p ^ = p 0 ( 1 − p 0 ) n = 0.30 ( 1 − 0.30 ) 300 = 0.02646 Any statistics that is below the critical value z 0.05 = − 1.645 will lead to reject the null hypothesis. So, reject H 0 if p ^ < p 0 − z α . p 0 ( 1 − p 0 ) n < 0.30 − 1.645 × 0.02646 < 0.257 Section 1: If the correct population proportion is 0.2, then average of the sampling distribution μ p ^ = 0.2 and standard deviation is σ p ^ = p 0 ( 1 − p 0 ) n = 0.2 ( 1 − 0.2 ) 300 = 0.02309 Now, calculate the probability of Type II error: β = P ( Type II error ) = P ( not rejecting H 0 when H 1 is true ) = P ( p ^ > 0.257 ) = P ( z > 0.257 − 0.2 0.2 ( 1 − 0.2 ) 300 ) = P ( z > 0.057 0.02309 ) = P ( z > 2.47 ) = 0.0068 The value of β is 0.0068. Probability of H 0 rejection when H 1 is true is 1 − β . The value 1 − β is referred to as the power of the test. The power of test is the probability that null hypothesis is correctly rejected. It is calculated as follows: 1 − β = 1 − 0.0068 = 0.9932 Hence, power of test is 0.9932. Section 2: If the correct population proportion is 0.22, then average of the sampling distribution μ p ^ = 0.22 and standard deviation is σ p ^ = p 0 ( 1 − p 0 ) n = 0.22 ( 1 − 0.22 ) 300 = 0.02398 Now, calculate the probability of Type II error: β = P ( Type II error ) = P ( not rejecting H 0 when H 1 is true ) = P ( p ^ > 0.257 ) = P ( z > 0.257 − 0.22 0.22 ( 1 − 0.22 ) 300 ) = P ( z > 0.037 0.02398 ) = P ( z > 1.55 ) = 0.0606 The value of β is 0.0606. Probability of H 0 rejection when H 1 is true is 1 − β . The value 1 − β is referred to as the power of the test. The power of test is the probability that null hypothesis is correctly rejected. It is calculated as follows: 1 − β = 1 − 0.0606 = 0.9394 Hence, power of test is 0.9394. Section 3: If the accurate population proportion is 0.24, then average of the sampling distribution μ p ^ = 0.24 and standard deviation is σ p ^ = p 0 ( 1 − p 0 ) n = 0.24 ( 1 − 0.24 ) 300 = 0.02466 Now, calculate the probability of Type II error: β = P ( Type II error ) = P ( not rejecting H 0 when H 1 is true ) = P ( p ^ > 0.257 ) = P ( z > 0.257 − 0.24 0.24 ( 1 − 0.24 ) 300 ) = P ( z > 0.017 0.02466 ) = P ( z > 0.69 ) = 0.2451 The value of β is 0.2451. Probability of H 0 rejection when H 1 is true is 1 − β . The value 1 − β is referred to as the power of the test. The power of test is the probability that null hypothesis is correctly rejected. It is calculated as follows: 1 − β = 1 − 0

Statistics: Informed Decisions Using Data (5th Edition)

5th Edition

ISBN: 9780134133539

Author: Michael Sullivan III

Publisher: PEARSON

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Chapter 10.6, Problem 17AYU

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Statistics: Informed Decisions Using Data (5th Edition)

Ch. 10.1 - 1. A ___________ is a statement regarding a...Ch. 10.1 - 2. ___________ ___________ is a procedure, based...Ch. 10.1 - 3. The ___________ ________ is a statement of no...Ch. 10.1 - 4. The ___________ ___________ is a statement we...Ch. 10.1 - 5. If we reject the null hypothesis when the...Ch. 10.1 - 6. If we do not reject the null hypothesis when...Ch. 10.1 - 7. The ___________ is the probability of making a...Ch. 10.1 - 8. True or False: Sample evidence can prove a null...Ch. 10.1 - Prob. 9AYU Ch. 10.1 - In Problems 9–14, the null and alternative...

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