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True or False: Sample evidence can prove a null hypothesis is true.

To test \(H_{0}: \sigma=1.2\) versus \(H_{1}: \sigma \neq 1.2,\) a random sample of size \(n=22\) is obtained from a population that is known to be normally distributed. (a) If the sample standard deviation is determined to be \(s=0.8\), compute the test statistic. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.10\) level of significance, determine the critical values. (c) Draw a chi-square distribution and depict the critical regions. (d) Will the researcher reject the null hypothesis? Why?

In Problems \(9-14,\) the null and alternative hypotheses are given. Determine whether the hypothesis test is lefi-tailed, right-tailed, or two-tailed. What parameter is being tested? \(H_{0}: p=0.2\) \(H_{1}: p<0.2\)

A machine fills bottles with 64 fluid ounces of liquid. The quality-control manager determines that the fill levels are normally distributed with a mean of 64 ounces and a standard deviation of 0.42 ounce. He has an engineer recalibrate the machine in an attempt to lower the standard deviation. After the recalibration, the quality-control manager randomly selects 19 bottles from the line and determines that the standard deviation is 0.38 ounce. Is there less variability in the filling machine? Use the \(\alpha=0.01\) level of significance.

In Problems \(9-14,\) the null and alternative hypotheses are given. Determine whether the hypothesis test is lefi-tailed, right-tailed, or two-tailed. What parameter is being tested? \(H_{0}: \sigma=4.2\) \(H_{1}: \sigma \neq 4.2\)

A manufacturer of high-strength, lowalloy steel beams requires that the standard deviation of yield strength not exceed 7000 pounds per square inch (psi). The quality-control manager selected a sample of 20 steel beams and measured their yield strength. The standard deviation of the sample was 7500 psi. Assume that yield strengths are normally distributed. Does the evidence suggest that the standard deviation of yield strength exceeds 7000 psi at the \(\alpha=0.01\) level of significance?

In Problems \(9-14,\) the null and alternative hypotheses are given. Determine whether the hypothesis test is lefi-tailed, right-tailed, or two-tailed. What parameter is being tested? \(H_{0}: p=0.76\) \(H_{1}: p>0.76\)

Ready for College? The ACT is a college entrance exam. ACT has determined that a score of 22 on the mathematics portion of the ACT suggests that a student is ready for college-level mathematics. To achieve this goal, ACT recommends that students take a core curriculum of math courses: Algebra I, Algebra II, and Geometry. Suppose a random sample of 200 students who completed this core set of courses results in a mean ACT math score of 22.6 with a standard deviation of \(3.9 .\) Do these results suggest that students who complete the core curriculum are ready for college-level mathematics? That is, are they scoring above 22 on the math portion of the ACT? (a) State the appropriate null and alternative hypotheses. (b) Verify that the requirements to perform the test using the \(t\) -distribution are satisfied. (c) Use the classical or \(P\) -value approach at the \(\alpha=0.05\) level of significance to test the hypotheses in part (a). (d) Write a conclusion based on your results to part (c).

Throwing darts at the stock pages to decide which companies to invest in could be a successful stock-picking strategy. Suppose a researcher decides to test this theory and randomly chooses 100 companies to invest in. After 1 year, 53 of the companies were considered winners; that is, they outperformed other companies in the same investment class. To assess whether the dart-picking strategy resulted in a majority of winners, the researcher tested \(H_{0}: p=0.5\) versus \(H_{1}: p>0.5\) and obtained a \(P\) -value of \(0.2743 .\) Explain what this \(P\) -value means and write a conclusion for the researcher.

In Problems \(9-14,\) the null and alternative hypotheses are given. Determine whether the hypothesis test is lefi-tailed, right-tailed, or two-tailed. What parameter is being tested? \(H_{0}: \mu=120\) \(H_{1}: \mu<120\)