Due Friday 4/17 at 10pm.

This assignment has 28 points possible. Your score will be scaled to be out of 5 points.

Question 1 – Sex Equality (4 pts)

The General Social Survey asked a random sample of 1,390 Americans the following question: “On the whole, do you think it should or should not be the government’s responsibility to promote equality between men and women?” 82% of the respondents said it “should be”. At a 95% confidence level, this sample has 2% margin of error. Based on this information, determine if the following statements are true or false, and explain your reasoning. (NORC 2016)

Part A: We are 95% confident that 80% to 84% of Americans in this sample think it’s the government’s responsibility to promote equality between men and women.

Part B: We are 95% confident that 80% to 84% of all Americans think it’s the government’s responsibility to promote equality between men and women.

Part C: If we considered many random samples of 1,390 Americans, and we calculated 95% confidence intervals for each, 95% of these intervals would include the true population proportion of Americans who think it’s the government’s responsibility to promote equality between men and women.

Part D: In order to decrease the margin of error to 1%, we would need to quadruple (multiply by 4) the sample size.

Question 2 – Proof of COVID-19 vaccination (6 pts)

In the US, businesses and schools shut down due to the COVID-19 pandemic in March 2020, and a vaccine became publicly available for the first time in April 2021. That month, a Gallup poll surveyed a random sample of 3,731 US adults, asking how they felt about the COVID-19 vaccine requirement for air travel. The poll found that 57% said they would favor it. (Gallup 2021b)

Part A: Describe the population parameter of interest.

Part C: Construct a 95% confidence interval for the proportion of US adults who favored requiring proof of COVID-19 vaccination for travel by airplane. (Show work and final value)

Part D: Interpret the confidence interval.

Part E: Without doing any calculations, describe what would happen to the confidence interval if we decided to use a higher confidence level.

Part F: Without doing any calculations, describe what would happen to the confidence interval if we used a larger sample.

Question 3 – Is Yawning Contagious? (3 pts)

Is yawning contagious? An experiment conducted by the MythBusters, a science entertainment TV program on the Discovery Channel, tested if a person can be subconsciously influenced into yawning if another person near them yawns. 50 people were randomly assigned to two groups: 34 to a group where a person near them yawned (treatment) and 16 to a group where there wasn’t a person yawning near them (control). The visualization below displays how many participants yawned in each group.

Suppose we are interested in estimating the difference in yawning rates between the control and treatment groups using a confidence interval. Explain why we cannot construct such an interval using the normal approximation. What might go wrong if we constructed the confidence interval despite this problem?

Question 4 – Sleep deprivation (5 pts)

A CDC report on sleep deprivation rates shows that the proportion of California residents who reported insufficient rest or sleep during each of the preceding 30 days is 8.0%, while this proportion is 8.8% for Oregon residents. These data are based on simple random samples of 11,545 California and 4,691 Oregon residents.

Construct a 90% confidence interval for the difference in proportions (Oregon - California) and interpret the interval in context.

Question 5 (Normal Stuff)

What percent of a standard normal distribution N\((\mu=0, \sigma=1)\) is found in each region denoted by a Z inequality below? Be sure to sketch a graph to double-check. (Hint: We previously used R to calculate normal probabilities. You might choose to use a different source, such as a Shiny App or a normal table. You do not need to turn in sketches)

Question 6 (CLT)

Define the term ‘sampling distribution’ of the sample proportion, and describe how the shape, center, and spread of this distribution change as sample size increases when p = 0.1.

Question 7 (Vegetarian Students)

Hint: Answer question 6 first.

Suppose that 8% of college students are vegetarians. Determine if the following statements are true or false, and explain your reasoning.

  1. The distribution of the sample proportions of vegetarians in random samples of size 60 is approximately normal.
  2. The distribution of the sample proportions of vegetarian college students in random samples of size 50 is right skewed.
  3. A random sample of 125 college students where 12% are vegetarians would be considered unusual.
  4. A random sample of 250 college students where 12% are vegetarians would be considered unusual.
  5. The standard error would be reduced by one-half if we increased the sample size from 125 to 250.

Question 8 (Legalization)

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 2022)

  1. Is 60% a sample statistic or a population parameter? Explain.
  2. Construct a 95% confidence interval for the proportion of US adults who think marijuana should be made legal, and interpret it in the context of the data.
  3. A critic points out that this 95% confidence interval is only accurate if the statistic follows a normal distribution, or if the normal model is a good approximation. Do the technical conditions hold for these data? Explain.
  4. A news piece on this survey’s findings states, “Majority of US adults think marijuana should be legalized.” Based on your confidence interval, is the news piece’s statement justified?

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