Question

Recently, the RMC manufacturing facility in Tupelo, Mississippi has been experiencing problems with one of their suppliers. This supplier provides a part for which the key quality characteristic is its weight. The problem has not been with the average weight. Instead, parts received at the Tupelo facility have been exhibiting excessive weight variability. Under normal conditions, the standard deviation in weight should be less than or equal to 4 g. To deal with this problem, a RMC quality engineer has decided to implement a test of hypotheses on each shipment of parts received at the facility. This test is to have a level of significance of 0.05 and can be summarized as follows where  denotes the standard deviation of part weight.

H0 : \sigma \leq 4g

H1 : \sigma > 4g

In designing this test, the engineer wishes to have a Type II error probability of 0.1 when the true standard deviation rises to 10 g

a. ) State any assumptions that are required to perform this test

b.) Determine the appropriate sample size for this test. You may use this sample size for the remainder of the problem.

c.) What is the power of this test if the true standard deviation of part weight is ? = 12 g?

Suppose a shipment is received, a random sample of parts is measured, and the resulting sample standard deviation is 5.5 g.

d.) Identify the test statistic and critical region for the test. State the conclusion for this test.

e.) Construct an appropriate confidence interval to verify your conclusion from part d.

f.) What is the P-value for this test? State the conclusion for this test. (An interval of probabilities is enough for the P-value)

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Solution

Part a: We assume that weight of a part is normally distributed. Part b: Minitab output: Power and Sample Size Test for One SPower Curve for One Standard Deviation 1.0 Sample Size 7 0.8 Assumptions Alpha 0.05 Alternative 0.6 Power 0.4 0.2 0.0 + 1.0 1

Sample Ratio Size Power 370.965915 Power Curve for One Standard Deviation 1.0 Sample Size 7 0.8 Assumptions Alpha 0.05 AlternAnswer: Required power=0.965915 Part d: 11.3438 Test statistic = x (n-1)s 42 where, n = 7, s= sample sd = 5.5. Critical valueStatistics N StDev Variance 75.50 30.3 95% Confidence Intervals Cl for Cl for Method StDev Variance Chi-Square (3.54, 12.11)Statistics N StDev Variance 75.50 30.3 95% One-Sided Confidence Intervals Lower Bound for Lower Bound Method StDev for Varian

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