Question

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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 o denotes the standard deviation of part weight He: 0 s 4g Hi: 0 > 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.

Answer 1

Sol:

a)

Sometimes hypothesis tests on the population variance or standard deviation are needed. When the population is modeled by a normal distribution, the tests and intervals described below are applicable. Whereas tests on means are relatively insensitive to the normality assumption, test procedure for variance are not.

b)

Suppose that we wish to test the hypothesis that the variance of a normal population $\sigma$ ² equals a specified value $\sigma$ ₀², say or equivalently, that the standard deviation $\sigma$ is equal to $\sigma$ ₀.

the hypothesized value of the standard deviation is less than or equal to 4g . Suppose that if the true standard deviation rises to 12g. Type II error probability is equal to 0.1.

$\lambda$ = 12g / 4g =3.

level of significance = 0.05

From the operating characteristic curve with $\beta$ = 0.20 and $\lambda$ = 0.1, we find that sample size (n) = 15,

c)

The power of a statistical test is the probability of rejecting the null hypothesis H₀ when the alternative hypothesis is true. The power of a statistical test = 1- $\beta$ =0.8 .