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What effect does increasing the sample size have on the power of the test, assuming all else remains unchanged?

Suppose you wish to determine if the mean IQ of students on your campus is different from the mean IQ in the general population, \(100 .\) To conduct this study, you obtain a simple random sample of 50 students on your campus, administer an IQ test, and record the results. The mean IQ of the sample of 50 students is found to be 107.3 with a standard deviation of \(13.6 .\) (a) Conduct a hypothesis test (preferably using technology) \(H_{0}: \mu=\mu_{0}\) versus \(H_{1}: \mu \neq \mu_{0}\) for \(\mu_{0}=103,104,105,106,107,108,109,110,111,112\) at the \(\alpha=0.05\) level of significance. For which values of \(\mu_{0}\) do you not reject the null hypothesis? (b) Construct a \(95 \%\) confidence interval for the mean IQ of students on your campus. What might you conclude about how the lower and upper bounds of a confidence interval relate to the values for which the null hypothesis is rejected? (c) Suppose you changed the level of significance in conducting the hypothesis test to \(\alpha=0.01\). What would happen to the range of values of \(\mu_{0}\) for which the null hypothesis is not rejected? Why does this make sense?

In \(2000,58 \%\) of females aged 15 and older lived alone, according to the U.S. Census Bureau. A sociologist tests whether this percentage is different today by conducting a random sample of 500 females aged 15 and older and finds that 285 are living alone. Is there sufficient evidence at the \(\alpha=0.1\) level of significance to conclude the proportion has changed since \(2000 ?\)

From Super Bowl I (1967) through Super Bowl XXXI (1997), the stock market increased if an NFL team won the Super Bowl and decreased if an AFL team won. This condition held 28 out of 31 years. (a) Suppose the likelihood of predicting the direction of the stock market (increasing or decreasing) in any given year is \(0.50 .\) Decide on the appropriate null and alternative hypotheses to test whether the outcome of the Super Bowl can be used to predict the direction of the stock market. (b) Use the binomial probability distribution to determine the \(P\) -value for the hypothesis test from part (a). (c) Comment on the dangers of using the outcome of the hypothesis test to judge investments. Be sure your comment includes a discussion of circumstances in which associations have a causal relationship.

Simulation Simulate drawing 100 simple random samples of size \(n=15\) from a population that is normally distributed with mean 100 and standard deviation 15 . (a) Test the null hypothesis \(H_{0}: \mu=100\) versus \(H_{1}: \mu \neq 100\) for each of the 100 simple random samples. (b) If we test this hypothesis at the \(\alpha=0.05\) level of significance, how many of the 100 samples would you expect to result in a Type I error? (c) Count the number of samples that lead to a rejection of the null hypothesis. Is it close to the expected value determined in part (b)? (d) Describe how we know that a rejection of the null hypothesis results in making a Type I error in this situation.

A can of soda is labeled as containing 12 fluid ounces. The quality control manager wants to verify that the filling machine is neither over-filling nor under-filling the cans. (a) Determine the null and alternative hypotheses that would be used to determine if the filling machine is calibrated correctly. (b) The quality control manager obtains a sample of 75 cans and measures the contents. The sample evidence leads the manager to reject the null hypothesis. Write a conclusion for this hypothesis test. (c) Suppose, in fact, the machine is not out of calibration. Has a Type I or Type II error been made? (d) Management has informed the quality control department that it does not want to shut down the filling machine unless the evidence is overwhelming that the machine is out of calibration. What level of significance would you recommend the quality control manager use? Explain.

Reading at Bedtime It is well-documented that watching TV, working on a computer, or any other activity involving artificial light can be harmful to sleep patterns. Researchers wanted to determine if the artificial light from e-Readers also disrupted sleep. In the study, 12 young adults were given either an iPad or printed book for four hours before bedtime. Then, they switched reading devices. Whether the individual received the iPad or book first was determined randomly. Bedtime was \(10 \mathrm{P.M}\). and the time to fall asleep was measured each evening. It was found that participants took an average of 10 minutes longer to fall asleep after reading on an iPad. The \(P\) -value for the test was \(0.009 .\) (a) What is the research objective? (b) What is the response variable? It is quantitative or qualitative? (c) What is the treatment? (d) Is this a designed experiment or observational study? What type? (e) The null hypothesis for this test would be that there is no difference in time to fall asleep with an e-Reader and printed book. The alternative is that there is a difference. Interpret the \(P\) -value.

According to the Centers for Disease Control, \(15.2 \%\) of American adults experience migraine headaches. Stress is a major contributor to the frequency and intensity of headaches. A massage therapist feels that she has a technique that can reduce the frequency and intensity of migraine headaches. (a) Determine the null and alternative hypotheses that would be used to test the effectiveness of the massage therapist's techniques. (b) A sample of 500 American adults who participated in the massage therapist's program results in data that indicate that the null hypothesis should be rejected. Provide a statement that supports the massage therapist's program. (c) Suppose, in fact, that the percentage of patients in the program who experience migraine headaches is \(15.3 \%\). Was a Type I or Type II error committed?

Based on historical birthing records, the proportion of males born worldwide is \(0.51 .\) In other words, the commonly held belief that boys are just as likely as girls is false. Systematic lupus erythematosus (SLE), or lupus for short, is a disease in which one's immune system attacks healthy cells and tissue by mistake. It is well known that lupus tends to exist more in females than in males Researchers wondered, however, if families with a child who had lupus had a lower ratio of males to females than the general population. If this were true, it would suggest that something happens during conception that causes males to be conceived at a lower rate when the SLE gene is present. To determine if this hypothesis is true, the researchers obtained records of families with a child who had SLE A total of 23 males and 79 females were found to have SLE. The 23 males with SL \(E\) had \(a\) total of 23 male siblings and 22 female siblings The 79 females with SLE had a total of 69 male siblings and 80 female siblings Source L.N. Moorthy, M.G.E. Peterson, K.B. Onel, and T.J.A. Lehman. "Do Children with Lupus Have Fewer Male Siblings" Laprus 2008 \(17: 128-131,2008\) (a) Explain why this is an observational study. (b) Is the study retrospective or prospective? Why? (c) There are a total of \(23+69=92\) male siblings in the study How many female siblings are in the study? (d) Draw a relative frequency bar graph of gender of the siblings. (e) Find a point estimate for the proportion of male siblings in families where one of the children has SLE (f) Does the sample evidence suggest that the proportion of male siblings in families where one of the children has SLE is less than 0.51 , the accepted proportion of males born in the general population? Use the \(\alpha=0.05\) level of significance. (g) Construct a \(95 \%\) confidence interval for the proportion of male siblings in a family where one of the children has SLE.

Explain the difference between statistical significance and practical significance.