The following problems are copied from the chapter 16 and Chapter 12 exercises from Introduction to Modern Statistics First Edition by Mine Çetinkaya-Rundel and Johanna Hardin (https://openintro-ims.netlify.app/inference-one-prop.html)

Show as much work as possible. Indicate what values you use, n =, z =, p =, when appropriate.

Chapter 16:

  1. CLT for proportions. Define the term “sampling distribution” of the sample proportion, and describe how the shape, center, and spread of the sampling distribution change as the sample size increases when p = .1. (Use StatKey with sample sizes of 10, 25, 50, and 100 to see the effect of sample size on shape, center, and spread.) Answer: The sampling distribution is the distribution of the random variable whose values contain all possible sample proportions. As the sample size increases the shape becomes more normal/Bell Shape with a center more lined up at the center of the Bell as larger samples are used. The spread also decreases with a larger sample.

20 Proof of COVID-19 vaccination. A Gallup poll surveyed 3,731 randomly sampled US in April 2021, asking how they felt about requiring proof of COVID-19 vaccination for travel by airplane. The poll found that 57% said they would favor it. (Gallup 2021b)

a Describe the population parameter of interest. What is the value of the point estimate of this parameter? Answer:The population parameter of interest are the 3731 randomly sampled US citizens in April 2021 about proof of Covid-19 vaccination to board a plane. The point estimate for those favoring it is .57.

  1. Construct a 95% confidence interval for the proportion of US adults who favor requiring proof of COVID-19 vaccination for travel by airplane. Answer: .57 + or - 2*.081=Confidence Interval. (.408, .732). I am 95% Confident that the proportion of US adults who favor requiring proof of Covid-19 Vaccination for travel by airplance lies within the interval (.408, .732)

  2. Without doing any calculations, describe what would happen to the confidence interval if we decided to use a higher confidence level. Answer: The width would increase as the standard error would get multiplied by a larger number.

  3. Without doing any calculations, describe what would happen to the confidence interval if we used a larger sample. Answer:The spread would decrease as the standard error would be less.

  1. Legalization of marijuana, mathematical interval. The General Social Survey asked a random sample of 1,563 US adults: Do you think the use of marijuana should be made legal, or not? 60% of the respondents said it should be made legal. (NORC 2018)
  1. Is 60% a sample statistic or a population parameter? Explain.

ANSWER: A sample statistic as it the 60% value is a measure for the sample, not the population as a whole.

  1. Using boostrapping to construct a 95% confidence interval for the proportion of US adults who think marijuana should be made legal, and interpret it.

ANSWER: 95% Confidence Interval: .6 + or - 2*.012= (.576, .624). I am 95% Confident that the proportion of US adults who believe that marijuana should be legal lies within the interval (.576, .624)

  1. A news piece on this survey’s findings states, “Majority of US adults think marijuana should be legalized.” Based on your confidence interval, is this statement justified?

ANSWER: Yes as the width of the interval is above 50% on the low end through the high end of the spread and therefore we are 95% confident that the majority of the U.S. wants marijuana to be legalized.

  1. Coupons driving visits. A store randomly samples 603 shoppers over the course of a year and finds that 142 of them made their visit because of a coupon they’d received in the mail. Use bootstrapping to construct a 95% confidence interval for the fraction of all shoppers during the year whose visit was because of a coupon they’d received in the mail.

ANSWER: .235 + or - 2*.017= (.203, .271). I am 95% confident that the proportion of shoppers who visit a certain shop because they received a coupon in the mail lies in the interval (.203, .271).

Chapter 12

  1. Waiting at an ER. A 95% confidence interval for the mean waiting time at an emergency room (ER) of (128 minutes, 147 minutes). Answer the following questions based on this interval.
  1. A local newspaper claims that the average waiting time at this ER exceeds 3 hours. Is this claim supported by the confidence interval? Explain your reasoning.

ANSWER: No as the interval shows that we only have a 95% confidence in the range of 128 minutes to 147 minutes both of which are lower than 180.

  1. The Dean of Medicine at this hospital claims the average wait time is 2.2 hours. Is this claim supported by the confidence interval? Explain your reasoning.

ANSWER: This could be correct as his claim falls within our confidence interval, we do not know for sure if that is the exact average but it has a high chance of being correct as determined by our 95% confidence interval.

  1. Without actually calculating the interval, determine if the claim of the Dean from part (b) would be supported based on a 99% confidence interval?

ANSWER: Yes as the width is larger in a 99% confidence interval so that we can be more certain that the average time falls within our interval.

Date and time completed: Tue Oct 4 17:03:37 2022