Due Friday 11/08 at 10pm.

This assignment has 25 points possible. Your score will be calculated out of 22 points and 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.

knitr::include_graphics("https://nfriedrichsen.github.io/homework/mythbusters-chart.jpg")

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 – Rainbow Trout (7 pts)

Rainbow trout were captured using 2 different types of nets; Sinking Net and Floating Net. It is hypothesized that the Sinking Net will be more effective in catching smaller fish. Out of all of the rainbow trout caught, a random sample of 71 fish caught using the sinking net was examined and a random sample of 52 fish caught using the floating net was examined.

Research Question: Estimate the difference in pop. mean length (mm) for fish caught with the floating net vs. the sinking net

Sinking Net Fish

The average length (mm) for the fish caught in the Sinking Net was 254.972 with a standard deviation of 73.619.

Floating Net Fish

The average length (mm) for the fish caught in the Floating Net was 278.846 with a standard deviation of 70.653.

Part A: Are the conditions met to make a 90% CI for difference in means for these samples? Explain.

Part B: Regardless of your answer to Part A, create a 90% CI for the difference in pop. means. (show work for credit). The \(t^*\) value needed is

qt(.95, df=51)
## [1] 1.675285

Part C: Interpret the confidence interval in context.

Part D: Is it plausible there is no difference in the pop. means according to the CI?