Exam 4: Inference From Data: Principles

a. Suppose Alejandro studies a random sample of data on a categorical variable and calculates a $95 \%$ confidence interval for the population proportion to be (.546,.674). Determine what the sample proportion must have been, and explain why. b. Suppose Brad and Carly plan to collect separate random samples, with Brad using a sample size of 500 and Carly using a sample size of 1500 . If Brad plans to construct a $99 \%$ confidence interval for the population proportion and Carly plans to construct a $90 \%$ confidence interval, who is more likely to obtain an interval that succeeds in capturing the population proportion? Explain. c. Suppose three students conduct a group project to estimate the proportion of students at their university who are from a different state. They take a random sample of students and find that $20 \%$ of their sample of students are from a different state. They then determine the following confidence intervals for the population proportion who are from a different state. One of these is a $90 \%$ confidence interval, one is a $99 \%$ confidence interval, and one is incorrect. Identify which is which. (In other words, write " $90 \%$ " below the appropriate interval, "99\%" below another, and "incorrect" below the third.) $(.116, .284)$ $(.176, .344)$ $(.146, .254)$

In the August 12, 2007, issue of Parade magazine (which comes with the Sunday newspaper for millions of Americans), readers were asked to go online and vote on this question: Should the drinking age be lowered? The results were published in the October 7 issue; more than 14,000 readers voted, and 48\% said "yes." -Use these sample data to determine a $99 \%$ confidence interval for the population proportion who favor lowering the drinking age.

Students enrolled in an introductory statistics course at a university were asked to take a survey that indicated whether the student had 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. Treat these students as a random sample of students at this university. -Describe what a Type II error means in this situation.

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 . -Determine the $p$ -value of the test as accurately as possible.

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. -State the relevant null and alternative hypotheses.

Students in an introductory statistics class were asked to report the age of their mothers when they were born. Summary statistics include Sample size: 28 students Sample mean: 29.643 years Sample standard deviation: 4.564 years a. Calculate the standard error of this sample mean. b. Determine and interpret a 90\% confidence interval for the mother's mean age (at student's birth) in the population of all students at this university. c. How would a $99 \%$ confidence interval compare to the $90 \%$ interval in terms of its midpoint and halfwidth? d. Would you expect $90 \%$ of the ages in the sample to be within the $90 \%$ confidence interval? Explain why or why not. e. Even if the distribution of mothers' ages were somewhat skewed, would this confidence interval procedure still be valid with these data? Explain why or why not.

Students enrolled in an introductory statistics course at a university were asked to take a survey that indicated whether the student had 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. Treat these students as a random sample of students at this university. -Describe the conclusion that you would reach at the $\alpha=.10$ significance level.

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 $\hat{p}$ stand for?

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. -How would a $99 \%$ confidence interval compare to this one in terms of its midpoint and half-width? (Do not bother to determine this interval.)

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. -Check whether the technical condition concerning sample size is satisfied here.

In the August 12, 2007, issue of Parade magazine (which comes with the Sunday newspaper for millions of Americans), readers were asked to go online and vote on this question: Should the drinking age be lowered? The results were published in the October 7 issue; more than 14,000 readers voted, and 48\% said "yes." -Does Parade's sampling method give you any reason to doubt that the confidence interval in "sample data to determine a $99 \%$ confidence interval for the population proportion who favor lowering the drinking age."? Explain.

Students enrolled in an introductory statistics course at a university were asked to take a survey that indicated whether the student had 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. Treat these students as a random sample of students at this university. -Describe what a Type I error means in this situation.

Suppose you analyze data to assess whether the proportion of heart transplant deaths at St. George's Hospital in London significantly exceeded the national benchmark rate of $15 \%$ . a. Write a sentence describing what committing a Type I error would mean in this study. b. Write a sentence describing what committing a Type II error would mean in this study.

If the instructor suspects that most students at the university are visual learners, state the null and alternative hypotheses (in symbols and in words) to be tested.

In the August 12, 2007, issue of Parade magazine (which comes with the Sunday newspaper for millions of Americans), readers were asked to go online and vote on this question: Should the drinking age be lowered? The results were published in the October 7 issue; more than 14,000 readers voted, and 48\% said "yes." -Explain why this interval is so narrow.

Reconsider the previous problem about the age of a college student's mother when the student was born. Now suppose you want to conduct a significance test of whether the sample data provide strong evidence that the population mean mother's age is less than 30 years. a. State the appropriate null and alternative hypotheses in symbols. b. Calculate the test statistic. c. Determine (as accurately as possible) the $p$ -value of the test. d. State your test decision at the $\alpha=.10$ significance level. e. Summarize your conclusion in context, and explain the reasoning process by which you reach this conclusion. f. Would the $p$ -value have been larger, smaller, or the same if a larger sample size had been used and all else had turned out the same? (Do not bother to explain.) g. Would the $p$ -value have been larger, smaller, or the same if the sample standard deviation had been smaller and all else had turned out the same? (Do not bother to explain.) h. Would the $p$ -value have been larger, smaller, or the same if the sample mean had been smaller, and all else had turned out the same? (Do not bother to explain.)

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 . -Comment on whether the technical conditions of this $t$ -test are satisfied.

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 standard deviation of 3 inches or a sample standard deviation of 5 inches? Explain.

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 . -State the null and alternative hypotheses for testing whether the mean score (among all students at this university) differs from 0 .

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 $\$ 85.3$ . -Describe what the population mean $\mu$ means in this context.