4. H0: = 120 and Ha: = 120 are used to test whether a bath soap production proce

Question

4. H0: = 120 and Ha: = 120 are used to test whether a bath soap production process is meeting thestandard output of 120 bars per batch. Use a .05 level of significance for the test and a planning value of 5for the standard deviation.

a. If the mean output drops to 117 bars per batch, the firm wants to have a 98% chance of concluding thatthe standard production output is not being met. How large a sample should be selected?

b. With your sample size from part (a), what is the probability of concluding that the process is operatingsatisfactorily for each of the following actual mean outputs: 117, 118, 119, 121, 122, and 123 bars perbatch? That is, what is the probability of a Type II error in each case?

(Please show the steps so I can understand this, thank you!)

Explanation / Answer

Solution:

a. Null Hypothesis (Ho): µ = 120

Alternative Hypothesis (Ha): µ 120

Using Z-tables, the critical value is

= 1 – 0.95 = 0.05 and = 1 – 0.98 = 0.02

Z (0.05/2) = Z (0.025) = 1.96

Z (0.02) = 2.05

The sample size is given by:-

N = [Z (a/2) + Z ()]^2 ^2/(µ0 - µa)^2

N = [(1.96 + 2.05)]^2 (5)^2/(120 – 117)^2

N = 44.7

N = 45

b. Using Z-tables, the critical value is

= 1 – 0.95 = 0.05

Z (0.05/2) = Z (0.025) = 1.96

Reject Ho if Z -1.96 or if Z 1.96

Test Statistics

Z = (X-bar - µ)/ (/n)

Z = (X-bar – 120)/ (5/45)

Solve for X-bar, we get

At Z = -1.96, Z = 118.54

At Z = 1.96, Z = 121.46

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