Assignment:
1. A poll conducted in 2013 found that 52% of U.S. adult Twitter users get at least some news on Twitter, and the standard error for this estimate was 2.4%. Identify whether each of the following statements are true or false. Provide an explanation to justify each of your answers.
(a) The data provide statistically significant evidence that more than half of U.S. adult Twitter users get some news through Twitter using a significance level of a = 0.01.
(b) Since the standard error is 2.4%, we can conclude that 97.6% of all U.S. adult Twitter users were included in the study.
(c) If we want to reduce the standard error of the estimate, we should collect less data.
(d) If we construct a 90% confidence interval for the percentage of U.S. adults Twitter users who get some news through Twitter, this confidence interval will be wider than a corresponding 99% confidence interval.
2. Suppose a 95% confidence interval for the average waiting time at a hospital's emergency room (ER) is (128 minutes, 147 minutes).
(a) A local newspaper claims that the average waiting time at this ER exceeds 3 hours. What do you think of this claim? Explain your reasoning.
(b) The Dean of Medicine at this hospital claims the average wait time is 2.2 hours. What do you think of this claim? Explain your reasoning.
(c) Without actually calculating the interval, determine if the claim of the Dean from part (b) would be considered reasonable based on a 99% confidence interval? Explain your reasoning.
3. Gilfoyle is a contestant on the game show "Let's Make a Deal". The host has given Gilfoyle $500. In front of Gilfoyle are three curtains. Behind one curtain is a jar of peanut butter worth $3. Behind another curtain is a sewing machine worth $300. Behind the third curtain is a silver-plated tuba worth $1,200. Gilfoyle does not know which prize is behind which curtain. Suppose Gilfoyle is interested only in the cash value of the prizes (and is risk-neutral). Is Gilfoyle willing to trade the $500 for a chance to pick a curtain at random and take whatever prize it conceals? Explain by writing down the appropriate expected value calculation. publ ic administrat ion psma 6002 spr ing 2018.
4. Fueleconomy.gov, the official US government source for fuel economy information, allows users to share gas mileage information on their vehicles. The histogram below shows the distribution of gas mileage in miles per gallon (MPG) from 14 users who drive a 2012 Toyota Prius. The sample mean is 53.3 MPG and the standard deviation is 5.2 MPG. Note that these data are user estimates and since the source data cannot be verified, the accuracy of these estimates are not guaranteed. Mileage (in MPG) 40 45 50 55 60 65 0 2 4 6
(a) We would like to use these data to evaluate the average gas mileage of all 2012 Prius drivers. Do you think this is reasonable? Why or why not?
(b) The EPA claims that a 2012 Prius gets 50 MPG (city and highway mileage combined). Do these data provide strong evidence against this estimate for drivers who participate on fueleconomy.gov? Note any assumptions you must make as you proceed with the test.
(c) Calculate a 95% confidence interval for the average gas mileage of a 2012 Prius by drivers who participate on fueleconomy.gov.
5. The 2010 General Social Survey asked the question: "For how many days during the past 30 days was your mental health, which includes stress, depression, and problems with emotions, not good?" Based on responses from 1,151 US residents, the survey reported a 95% confidence interval of 3.40 to 4.24 days in 2010.
(a) What does "95% confidence interval" mean? Explain in the context of this question.
(b) Suppose the researchers think a 99% confidence level would be more appropriate for this interval. Will this new interval be smaller or larger than the 95% confidence interval?
(c) If a new survey were to be done with 500 Americans, would the standard error of the estimate be larger, smaller, or about the same? Why?