1. A technology company is interested in determining the effectiveness of its new advertising campaign in a specific market segment. The company believes that at least 36% of the individuals in this market segment will recall seeing the company's advertisements. As part of their evaluation, they surveyed a random sample of 200 individuals from this market segment and found that 85 of them recalled seeing the company's advertisements. At the 1% significance level, conduct a hypothesis test with the goal of proving his claim.

a. Specify the null hypothesis and the alternative hypothesis. State whether you are using a one- or two-tailed test and explain your reasoning.

b. Conduct calculations by hand to compute the critical value and p-value of the test. Determine your decision using two methods, the rejection region and the p-value approach.

c. Use R to conduct the hypothesis test. Copy and paste the command and report generated by R.

d. Communicate the results of the test in "plain English" including the meaning of the p-value.

2. A researcher in Washington, D.C. wants to estimate the mean number of minutes adult residents spend exercising. He takes a random sample of 70 adult residents and records their average time spent exercising as 72 minutes. Assume that the population standard deviation is 89 minutes.

a. Construct a 95% confidence interval for the population mean exercise time of adult residents in Washington, D.C. Use by hand calculations and R commands to build the confidence interval. For by hand calculations, use R to compute the z-multiple. Provide the step calculations and commands used in R. State the confidence interval in the two forms: point estimate margin of errorand [lower limit, upper limit].

b. Interpret the calculated 95% confidence interval within the context of the problem.

c. At 99% confidence, what is the margin of error?

d. What happens to the margin of error as we increase our confidence level from 95% to 99%?