Exam 4: Inference From Data: Principles

Suppose you measure the heights of a random sample of chief executive officers (CEOs) in order to study whether CEOs tend to be taller than the national average height of 69 inches. -If the sample mean turned out to be 70 inches, which would give a smaller $p$ -value: a sample size of 50 CEOs or a sample size of 200 CEOs?

Students enrolled in an introductory statistics course at a university were asked to take a survey that indicated whether the student's learning style was more visual or verbal. Each student received a numerical score ranging from -11 to +11 . Negative scores indicated a visual learner, and positive scores indicated a verbal learner. The closer the score was to -11 or +11 , the stronger the student's inclination toward that learning style. A score of 0 would indicate neutrality between visual or verbal learning. For the 39 students who took the survey, the mean score was -2.744 , and the standard deviation was 4.988 . -Calculate the value of the $t$ -test statistic for the hypotheses stated in " State the null and alternative hypotheses for testing whether the mean score (among all students at this university) differs from 0" .

a. Suppose you conduct a significance test and decide to reject the null hypothesis $\left(\mathrm{H}_{0}\right)$ at the $\alpha=.05$ level. If you conduct the same test on the same data but instead use the $\alpha=.01$ level, what decision would you make? (Select your answer. Do not bother to explain.) -Reject $\mathrm{H}_{0}$ -Fail to reject $\mathrm{H}_{0}$ -Cannot say without more information b. Suppose you conduct a significance test and decide to fail to reject the null hypothesis $\left(\mathrm{H}_{0}\right)$ at the $\alpha=.05$ level. If you conduct the same test on the same data but instead use the $\alpha=.01$ level, what decision would you make? (Select your answer. Do not bother to explain.) -Reject $\mathrm{H}_{0}$ -Fail to reject $\mathrm{H}_{0}$ -Cannot say without more information c. Suppose that we tell you that we flipped a coin multiple times and it landed heads $75 \%$ of the time. Would you be reasonably convinced that this was not a fair coin (where "fair" means that the coin has a .5 probability of landing "heads")? If so, explain why. If not, describe what additional information you would ask for and explain why it is necessary.

Students in an introductory statistics class were asked how many states they have visited. The following output pertains to the sample results: a. Determine a $90 \%$ confidence interval for the population mean number of states visited among all students at this university. b. Check and comment on whether the technical conditions of this confidence interval are satisfied. c. For what proportion of students in the sample is the number of states visited within the interval calculated in part a? d. Should you expect your answer to part $\mathrm{c}$ to be close to $90 \%$ ? Explain why or why not. e. Based on your interval, what can you say about the p-value if you were to conduct a two-sided significance test of whether the population mean differs from 15? Explain briefly, without conducting a test or doing new calculations.

You have learned that a confidence interval for is given by: $\overline{x} \pm t^*\frac{s}{\sqrt{n}^\cdot}$ . -What does the symbol $\mu$ stand for?

The following data are monthly rents (in dollars) of studio and one-bedroom apartments in Harrisburg, Pennsylvania, in 2007, obtained from a random sample of such apartments advertised at {rent.com} in July 2007: $500,549,569,575,585,600,630,680,705,790$ The mean of these ten rent prices is $\$ 618.3$ , and the standard deviation is $\$ 85.3$ . -Determine a $95 \%$ confidence interval for $\mu$ .

Students enrolled in an introductory statistics course at a university were asked to take a survey that indicated whether the student has a visual or verbal learning style. Of the 39 students who took the survey, 25 were judged to have a visual learning style and 14 were considered verbal learners. -Determine a $90 \%$ confidence interval for the population proportion who are visual learners at this university.

You have learned that a confidence interval for is given by: $\overline{x} \pm t^*\frac{s}{\sqrt{n}^\cdot}$ . -What does the symbol $s$ stand for?

You have learned that a confidence interval for $\pi$ is given by: $\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ . -What does the symbol $n$ stand for?

You have learned that a confidence interval for $\pi$ is given by: $\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ . -Does increasing the sample size lead to a larger value of $z^{*}$ , a smaller value of $z^{*}$ , or no change in the value of $z^{*}$ ?

Students enrolled in an introductory statistics course at a university were asked to take a survey that indicated whether the student has a visual or verbal learning style. Of the 39 students who took the survey, 25 were judged to have a visual learning style, and 14 were considered verbal learners. Consider these 39 students to be a random sample of students at this university. Conduct a test of significance of whether these sample data provide strong evidence that more than half of all students at the university have a visual learning style. Be sure to report the hypotheses, test statistic, and $p$ -value. Also check the technical conditions of the procedure, and write a conclusion in context.

Will you and your friend obtain different values of the test statistic? Explain briefly.

Students enrolled in an introductory statistics course at a university were asked to take a survey that indicated whether the student has a visual or verbal learning style. Of the 39 students who took the survey, 25 were judged to have a visual learning style and 14 were considered verbal learners. -Write a sentence interpreting what this interval says.

A statistics professor has asked his students to flip coins over the years. He has kept track of how many flips land heads and how many land tails. Combining the results of his students over many years, he has formed a $95 \%$ confidence interval for the long-run population proportion of heads to be $(.501, .513)$ . a. Why is this interval so narrow? b. Suppose he were to conduct a hypothesis test of whether the long-run population proportion of heads differs from one-half. Based on this interval (do not conduct the test), would he reject the null hypothesis at the $\alpha=.05$ significance level? Explain briefly (no more than one sentence). c. Does the interval provide strong evidence that the long-run population proportion of heads is much different from one-half? Explain briefly.

Will you and your friend obtain different values of the $p$ -value? Explain briefly. Suppose that a random sample of students at a university takes a survey that indicates whether the student has a visual or verbal learning style.

Students enrolled in an introductory statistics course at a university were asked to take a survey that indicated whether the student's learning style was more visual or verbal. Each student received a numerical score ranging from -11 to +11 . Negative scores indicated a visual learner, and positive scores indicated a verbal learner. The closer the score was to -11 or +11 , the stronger the student's inclination toward that learning style. A score of 0 would indicate neutrality between visual or verbal learning. For the 39 students who took the survey, the mean score was -2.744 , and the standard deviation was 4.988 . -Summarize the conclusion that you would draw from this test.

Suppose you want to estimate the proportion of people in your community who would choose Thin Mints as their favorite Girl Scout cookie. Also suppose previous studies have shown this proportion to be around .3 . a. Determine how many people you would have to sample in order to estimate this proportion to within $\pm.025$ with $90 \%$ confidence. If your boss says that this number (your answer to part a) is too large to be practical, you might respond that a smaller sample size would suffice if either the margin-of-error (.025) or the confidence level $(90 \%)$ were changed. b. In which direction would the margin-of-error need to change in order for a smaller sample size to suffice? c. In which direction would the confidence level need to change in order for a smaller sample size to suffice?