91Ó°ÊÓ

Pulse and Gender: Cl Using data from NHANES, we looked at the pulse rate for nearly 800 people to see whether it is plausible that men and women have the same population mean. NHANES data are random and independent. Minitab output follows.

Women's Heights Assume women's heights are approximately Normally distributed with a mean of 65 inches and a standard deviation of \(2.5\) inches. Which of the following questions can be answered using the Central Limit Theorem for sample means as needed? If the question can be answered, do so. If the question cannot be answered, explain why the Central Limit Theorem cannot be applied. a. Find the probability that a randomly selected woman is less than 63 inches tall. b. If five women are randomly selected, find the probability that the mean height of the sample is less than 63 inches. c. If 30 women are randomly selected, find the probability that the mean height of the sample is less than 63 inches.

Choose a test for each situation: one-sample \(t\) -test, two-sample \(t\) -test, paired \(t\) -test, and no \(t\) -test. a, A random sample of students who transfered to a 4 -year university from community colleges are asked their GPAs. Our goal is to determine whether the mean GPA for transfer students is significantly different from the population mean GPA for all students at the university. b. Students observe the number of office hours posted for a random sample of tenured and a random sample of untenured professors. c. A researcher goes to the parking lot at a large grocery chain and observes whether each person is male or female and whether they return the cart to the correct spot before leaving (yes or no).

Choose a \(t\) -test for each situation: one-sample \(t\) -test, twosample \(t\) -test, paired \(t\) -test, and no \(t\) -test. a. A random sample of car dealerships is obtained. Then a student walks onto each dealer's lot wearing old clothes and finds out how long it takes (in seconds) for a salesperson to approach the student. Later the student goes onto the same lot dressed very nicely and finds out how long it takes for a salesperson to approach. b. A researcher at a preschool selects a random sample of 4 -year-olds, determines whether they know the alphabet (yes or no), and records gender. c. A researcher calls the office phone for a random sample of faculty at a college late at night, measures the length of the outgoing message, and records gender.

Exam Grades The final exam grades for a sample of daytime statistics students and evening statistics students at one college are reported. The classes had the same instructor, covered the same material, and had similar exams. Using graphical and numerical summaries, write a brief description about how grades differ for these two groups. Then carry out a hypothesis test to determine whether the mean grades are significantly different for evening and daytime students. Assume that conditions for a \(t\) -test hold. Select your significance level. Daytime grades: \(100,100,93,76,86,72.5,82,63,59.5,53,79.5\), \(67,48,42.5,39\) Evening grades: \(100,98,95,91.5,104.5,94,86,84.5,73,92.5\), \(86.5,73.5,87,72.5,82,68.5,64.5,90.75,66.5\)

Why Is \(n-1\) in the Sample Standard Deviation? Why do we calculate \(s\) by dividing by \(n-1\), rather than just \(n\) ? $$ s^{2}=\frac{\sum(x-\bar{x})^{2}}{n-1} $$ The reason is that if we divide by \(n-1\), then \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\), the population variance. We want to show that \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\), sigma squared. The mathematical proof that this is true is beyond the scope of an introductory statistics course, but we can use an example to demonstrate that it is. First we will use a very small population that consists only of these three numbers: 1,2, and 5 . You can determine that the population standard deviation, \(\sigma\), for this population is \(1.699673\) (or about \(1.70\) ), as shown in the \(\mathrm{TI}-84\) output. So the population variance, sigma squared, \(\sigma^{2}\), is \(2.888889\) (or about 2.89). Now take all possible samples, with replacement, of size 2 from the population, and find the sample variance, \(s^{2}\), for each sample. This process is started for you in the table. Average these sample variances \(\left(s^{2}\right)\), and you should get approximately \(2.88889 .\) If you do. then you have demonstrated that \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\), sigma squared. Show your work by filling in the accompanying table and show the average of \(s^{2}\).