Exam 4: Inference From Data: Principles

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 the standard error of the sample mean.

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." -Without conducting a test of significance, what can you say about the $p$ -value if you were to test whether the population proportion differs from one-half? Explain.

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 confidence level lead to a larger value of $z^{*}$ , a smaller value of $z^{*}$ , or no change in the value of $z^{*}$ ?

It has been conjectured that two-thirds of all students have an active, as opposed to reflective, learning style. Data are available from a large sample of introductory statistics students at a public university in the eastern United States, who took a survey that assessed their learning style as active or reflective. Of the 962 students who participated, 596 were diagnosed as active learners and 366 as reflective learners. Use these sample data to conduct a significance test of the conjecture that two-thirds of all students have an active learning style. Report the hypotheses, test statistic, and $p$ -value. Include a check of technical conditions. Also indicate your test decision at the $\alpha=.05$ significance level, and summarize your conclusion in context.

You have learned that a confidence interval for is given by: $\overline{x} \pm t^*\frac{s}{\sqrt{n}^\cdot}$ . -Does increasing the sample size lead to a larger value of $t^{*}$ , a smaller value of $t^{*}$ , or no change in the value of $t^{*}$ ?

If the instructor has read about national results indicating that two-thirds of all students are visual learners, state the null and alternative hypotheses (in symbols and in words) to be tested in order to assess whether the same proportion holds for that university.

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 $\$ 85.3$ . -Would you expect about $95 \%$ of the studio and one-bedroom apartments in Harrisburg in July 2007 to have a rent price within this interval? Explain.

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. -Explain why you might feel wary about applying this confidence interval to the population of all students at this university.

The National Institutes of Health funded a study of a random sample of 856 people age 71 and older in the United States. They found that $9.7 \%$ of these people suffer from Alzheimer's Disease. a. Identify the population of interest in this study. b. Is .097 (the decimal version of $9.7 \%$ ) a parameter or a statistic? What symbol do you use to represent it? c. Determine a 95\% confidence interval for the proportion of elderly Americans who have Alzheimer's disease. d. Check and comment on whether the technical conditions required for this confidence interval are satisfied. e. If you were to determine instead a $90 \% \mathrm{CI}$ , how would it differ and how would it be similar to the result in part $\mathrm{c}$ ? (Do not bother to do the calculations.) f. Determine how many people would have to be studied in a new sample if you want to estimate the population proportion to within $\pm$ .02 with $99 \%$ confidence. (Use the result of the current sample in your determination of the new sample size.)

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 $\$ 85.3$ . -Write a sentence interpreting what this interval means.

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

Explain (briefly) what is wrong with each of the following sets of hypotheses: a. $\mathrm{H}_{0}:\pi=1.2, \quad \mathrm{H}_{\mathrm{a}}:\pi> 1.2$ b. $\mathrm{H}_{0}: \hat{ }p=.5, \quad \mathrm{H}_{\mathrm{a}}:\hat{p}> .5$ c. $\mathrm{H}_{0}:\mu=69, \quad \mathrm{H}_{\mathrm{a}}:\mu\geq 69$ d. $\mathrm{H}_{0}:\pi\neq .5, \quad \mathrm{H}_{\mathrm{a}}:\pi=.5$

Do hypotheses involve parameters or statistics? Suppose that you and a friend analyze the same sample data and conduct a test of significance about the same conjectured value, but you use a two-sided alternative and your friend uses a one-sided alternative.

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. -If the sample size had been ten times larger and the results had turned out proportionally identical, would the $p$ -value have been larger, smaller, or the same?

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." -Report the margin-of-error for this confidence interval.

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. -State the null and alternative hypotheses for testing whether the data provide strong evidence against the hypothesis that two-thirds of all students at the university are visual learners. The $p$ -value for this test turns out to be 734 .

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 all else (sample size and standard deviation) were the same, which would give stronger evidence that CEOs do tend to be taller than an average height of 69 inches: a sample mean of 70 inches or a sample mean of 71 inches? Explain.

A student wants to assess whether her dog Muffin tends to chase her blue ball more often than she chases her red ball. The student rolls both a blue ball and a red ball at the same time and observes which ball Muffin chooses to chase. Repeating this process a total of 96 times, the student finds that Muffin chased the blue ball 52 times and the red ball 44 times. a. What are the observational units in this study? b. What is the variable in this study? c. State the appropriate null and alternative hypotheses, in words and in symbols. d. Calculate the test statistic and $p$ -value. e. Would you reject the null hypothesis at the $\alpha=.10$ significance level? Explain. f. Write a one-sentence conclusion to the student, summarizing what the data reveal about whether her dog Muffin tends to chase her blue ball more often than her red ball. Include an explanation of what the $p$ value means in the context of this study.

You have learned that a confidence interval for is given by: $\overline{x} \pm t^*\frac{s}{\sqrt{n}^\cdot}$ . -Does increasing the confidence level lead to a larger value of $t^{*}$ , a smaller value of $t^{*}$ , or no change in the value of $t^{*}$ ?