problem 1: Mars Landing (no data set) A sample of 830 Americans was randomly selected on the population of all American adults. Among other questions, the sample was asked if they believe that the United States will land a human on Mars by 2050. Of those sampled, 544 stated that they believe this will happen.

a) Calculate the sample proportion of Americans who believe the US will land a human on Mars by 2050. Round this value to four decimal places.

b) Write one sentence each to check the three conditions of the Central Limit Theorem. Show your work for the mathematical check needed to show a large sample size was taken. c) Using the sample proportion obtained in (a), construct a 90% confidence interval to estimate the population proportion of Americans who believe the US will land a human on Mars by 2050. Please do this "by hand" using the formula and showing your work (please type your work, no images accepted here). Round your confidence limits to four decimal places. d) Verify your result from part

(c) using Stat Proportions Stats One Sample With Summary. Inside the box, enter the number of successes, the number of observations, select confidence interval, set it to the correct level, and click Compute! Copy and paste your StatCrunch result in your document. e) Interpret the StatCrunch confidence interval in part

(d) in one sentence using the context of the question.

f) Use the Confidence Interval applet (for a Proportion) in StatCrunch to simulate constructing one thousand 90% confidence intervals. Assume the population proportion of Americans who believe the US will land a human on Mars by 2050 is p = 0.62 and remember our sample size was n = 830. Once the window is open, click "Reset" and select (or click) 1000 intervals. Copy and paste your full image into your document.

g) Compare the "Prop. contained" value from part (f) to the confidence level associated with the simulation in one sentence.

h) Write a long-run interpretation for your confidence interval method in context in one sentence. Think about what would happen if you took many, many more samples