91Ó°ÊÓ

Determine the critical region and the critical values used to test the following null hypotheses: a. \(\quad H_{o}: \mu=55(\geq), H_{a}: \mu<55, \alpha=0.02\) b. \(\quad H_{o}: \mu=-86(\geq), H_{a}: \mu<-86, \alpha=0.01\) c. \(\quad H_{o}: \mu=107, H_{a}: \mu \neq 107, \alpha=0.05\) d. \(\quad H_{o}: \mu=17.4(\leq), H_{a}: \mu>17.4, \alpha=0.10\)

There are only two possible decisions as a result of a hypothesis test. a. State the two possible decisions. b. Describe the conditions that will lead to each of the two decisions identified in part a.

We are interested in estimating the mean life of a new product. How large a sample do we need to take to estimate the mean to within \(\frac{1}{10}\) of a standard deviation with \(90 \%\) confidence?

A manufacturing process produces ball bearings with diameters having a normal distribution and a standard deviation of \(\sigma=0.04 \mathrm{cm} .\) Ball bearings that have diameters that are too small or too large are undesirable. To test the null hypothesis that \(\mu=0.50 \mathrm{cm},\) a sample of 25 is randomly selected and the sample mean is found to be 0.51. a. Design null and alternative hypotheses such that rejection of the null hypothesis will imply that the ball bearings are undesirable. b. Using the decision rule established in part a, what is the \(p\) -value for the sample results? c. If the decision rule in part a is used with \(\alpha=0.02\) what is the critical value for the test statistic?

From a population of unknown mean \(\mu\) and a standard deviation \(\sigma=5.0,\) a sample of \(n=100\) is selected and the sample mean 41.5 is found. Compare the concepts of estimation and hypothesis testing by completing the following: a. Determine the \(95 \%\) confidence interval for \(\mu\) b. Complete the hypothesis test involving \(H_{a}: \mu \neq 40\) using the \(p\) -value approach and \(\alpha=0.05\) c. Complete the hypothesis test involving \(H_{a}: \mu \neq 40\) using the classical approach and \(\alpha=0.05\) d. On one sketch of the standard normal curve, locate the interval representing the confidence interval from part a; the \(z \star, p\) -value, and \(\alpha\) from part \(b ;\) and the \(z \\#\) and critical regions from part c. Describe the relationship between these three separate procedures.