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Question: For a certain wine, the mean pH (a measure of acidity) is supposed to be 3.35 with a known standa...

For a certain wine, the mean pH (a measure of acidity) is supposed to be 3.35 with a known standard deviation of σ = .13. The quality inspector examines 38 bottles at random to test whether the pH is too high, using a right-tailed test at α = 0.1.
  
(a) What is the power of this test if the true mean is μ = 3.37? (Round your Z-value to 3 decimal places and final answer to 4 decimal places.)

Power   

(b) What is the power of this test if the true mean is μ = 3.39? (Round your Z-value to 3 decimal places and final answer to 4 decimal places.)

Power

(c) What is the power of this test if the true mean is μ = 3.41? (Round your Z-value to 3 decimal places and final answer to 4 decimal places.)

a) p=0.1716 or 0.1715 is wrong

b)p=0.0289 is wrong

c)p=0.0022 is wrong

Answer 1

Power of the Test
Here, n = 38 , xbar = 3.35 and s = 0.13
Critical value is 1.2816
True mean = 3.37

Critical value, c = xbar + (z/t)*s/sqrt(n)
c = 3.35 + 1.2816 * 0.13/sqrt(38)
c = 3.377

Part A)

Beta or type II error is the probability of fail to reject the null hypothesis
P(X < 3.377 | mu = 3.37) = 0.63
beta = 0.63
Power of the test is 1 - beta
Power of the test is 0.37

Part B)

Beta or type II error is the probability of fail to reject the null hypothesis
P(X < 3.377 | mu = 3.39) = 0.2688
beta = 0.2688
Power of the test is 1 - beta
Power of the test is 0.7312

Part C)
Beta or type II error is the probability of fail to reject the null hypothesis
P(X < 3.377 | mu = 3.41) = 0.0588
beta = 0.0588
Power of the test is 1 - beta
Power of the test is 0.9412