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## Exercise 1

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 “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. Do not use a computer. You may only use $$+$$, $$-$$, $$\times$$, $$\div$$, and $$\sqrt{}$$ on a calculator. Show all work.

(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, $$\sigma$$.

(e) Construct a $$90\%$$ confidence upper bound for the true standard deviation of the weight of a “22 ounce” Charles Mulligan’s porterhouse steak, $$\sigma$$.

## Exercise 2

Last year, 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 marijuana legalization.

(b) Suppose that a pollster wants to estimate the true proportion of Champaign-Urbanians that support 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 marijuana legalization to within 0.04, with $$90\%$$ confidence. How many Champaign-Urbanians should this pollster poll?

## Exercise 3

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.

## Exercise 4

Suppose that 10 students visit the Stars Hollow Apple Orchard and each pick (a random sample of) 15 Fuji apples, weigh them, then create a $$90\%$$ confidence interval for the true mean weight of a Fuji apple at the Stars Hollow Apple Orchard. What is the probably that at most 2 of these intervals do not contain the true mean weight of a Fuji apple at the Stars Hollow Apple Orchard?