Please see the **detailed homework policy document** for information about homework formatting, submission, and grading.

Due to the nature of the exercises in this homework, instead of the usual 0 - 1 - 2 grading for each exercise, we will instead utilize 0 - 0.5 - 1 grading for each exercise **part**.

Before it closed, Ron Swanson was a frequent patron of Charles Mulligan’s Steakhouse in Indianapolis, Indiana. Ron enjoyed the experience so much, during each visit he took a picture with his steak.

Ron also weighed each steak he consumed. He has a record of eating six “22 ounce” Charles Mulligan’s porterhouse steaks. Ron found that these six steaks weighed

\[ 22.4 \text{ oz}, \ 20.8 \text{ oz}, \ 21.6 \text{ oz}, \ 20.2 \text{ oz}, \ 21.4 \text{ oz}, \ 22.0 \text{ oz} \]

Suppose that the weight of “22 ounce” Charles Mulligan’s porterhouse steaks follow a \(N(\mu, \sigma^2)\) distribution and that Ron’s six steaks were a random sample.

**(a)** Compute the sample standard deviation, \(s\), of these six steaks.

**(b)** Construct a \(95\%\) two-sided confidence interval for the true mean weight of a “22 ounce” Charles Mulligan’s porterhouse steak, \(\mu\).

**(c)** Construct a \(95\%\) confidence *lower bound* for the true mean weight of a “22 ounce” Charles Mulligan’s porterhouse steak, \(\mu\).

**(d)** Construct a \(90\%\) two-sided confidence interval for the true standard deviation of the weight of a “22 ounce” Charles Mulligan’s porterhouse steak, \(\sigma\).

In 2017, ballots in Champaign-Urbana contained the following question to assess public opinion on an issue:

“Should the State of Illinois legalize and regulate the sale and use of marijuana in a similar fashion as the State of Colorado?”

Suppose that we would like to understand Champaign-Urbana’s 2017 opinion on marijuana legalization. To satisfy our curiosity, we obtain a random sample of 120 Champaign-Urbanians and find that 87 support marijuana legalization.

**(a)** Construct a \(99\%\) confidence interval for \(p\), the true proportion of Champaign-Urbanians that support recreational marijuana legalization.

**(b)** Suppose that a pollster wants to estimate the true proportion of Champaign-Urbanians that support recreational marijuana legalization to within 0.04, with \(95\%\) confidence. How many Champaign-Urbanians should this pollster poll? Assume the pollster has no prior knowledge about the proportion.

**(c)** Now assume the pollster believes that support for legalization is somewhere between \(65\%\) and \(85\%\) and they would like to estimate the true proportion of Champaign-Urbanians that support recreational marijuana legalization to within 0.04, with \(90\%\) confidence. How many Champaign-Urbanians should this pollster poll?

**(d)** Instead, suppose we obtain a random sample of 80 Champaign voters, of which 55 support recreational marijuana legalization. We also obtain a random sample of 100 Urbana voters, of which 75 support recreational recreational marijuana legalization. Let \(p_C\) be the true proportion of Champaign voters who support recreational marijuana legalization and let \(p_U\) be the true proportion of Urbana voters who support recreational marijuana legalization. Calculate a 99% confidence interval for \(p_U - p_C\).

Suppose students in a Statistics class are interested in the average score of an exam, but the instructor has only graded (a random sample of) 13 of the (many) exams. The instructor states that a \(90\%\) confidence interval for the true mean is given by \((79.14, 82.86)\) and that you can assume the grades follow a normal distribution.

Using only this information, calculate \(\bar{x}\), \(s\), and finally, a \(95\%\) confidence interval for \(\mu\), the true mean of the exam.

Professor Professorson, a researcher at Greendale Community College, is interested in the effect of caffeine on the typing speed of students. Professorson obtains a random sample of 8 students who are given 400 mg of caffeine then given a typing test. (Don’t try this at home.) They type an average of 51.4 words per minute (wpm), with a sample standard deviation of 12.3 wpm. He also obtains a random sample of 13 students who are given a placebo before the typing test. The placebo group types an average of 43.9 wpm, with a sample standard deviation of 15.1 wpm. Assume typing speeds follow a normal distribution in both groups.

Construct a 99% confidence interval for \(\mu_C - \mu_P\), the true difference in average typing speed between the caffeine and placebo groups. *Assume that the two population variances are equal.*