1. The Normal and Standard Normal Distributions
Professor Economises has 362 students in her introductory economics class. Suppose that the scores on the final exam in this class are normally distributed with a mean of 75 points and a standard deviation of 10 points. How many students in the class can be expected to receive below 60 points? Convert the distribution to a standard normal distribution and express your answer to the nearest number of students. Show your work and explain.
2. Monte Carlo Simulation and the Central Limit Theorem in Excel
Suppose that a hot dog factory is attempting to produce foot-long hotdogs 12 inches in length. There is variation in the process, however, and the actual lengths of the "footlong" hotdogs vary uniformly between 11.5 and 12.5 inches in length. At regular intervals, a random sample of 10 hotdogs is selected, the lengths measured, and the average (mean) length of the ten hotdogs is recorded. After 500 samples of 10 hotdogs have been selected, the frequency distribution of the 500 sample means is plotted (the sampling distribution of the mean). Even though the data generating process (DGP) is a random uniform distribution and the sample size (n=10) is small, the distribution of average lengths tends toward the normal distribution. This Monte Carlo simulation should provide intuition regarding the Central Limit Theorem (CLT). Please follow the steps below to conduct this Monte Carlo simulation.
3. Applying the Central Limit Theorem
A nationwide study found that the average yearly claim per automobile insurance policy (after deductible) is $500 with a standard deviation of $2,500. Consider these parameter values to be the true population parameters.
Suppose that you are working for a small, start-up insurance company with only 10,000 automobile insurance policies, and the company has $4.5 million to pay for claims next year. What is the probability that the company will not have enough money set aside to pay off the claims next year? If the company does not have enough money set aside, it will declare bankruptcy and you will lose your job. Should you start working on your resume? Show your work and explain.
4. Hypothesis Testing by Hand
Consider the number of days absent from a random sample of seven students during a quarter in a large introductory economics class.
= -0, 1, 1, 3, 3, 8, 12.
Can you reject the null hypothesis that the population mean number of absences in this large introductory economics class is 8 at the ten-percent level of significance, at the 5- percent level of significance? Use the critical value approach. You can use GRETL for critical values, but you must show all of your calculations, using Equation Editor in Word, and explain.
5. Spending on Textbooks and Course Materials
Recently, it was claimed that college students spend an average of $1,200 per year on textbooks and course materials (see Ethan Senack, Fixing the Broken Textbook Market.
Assuming that the attached file (textbook.xls) is a random, representative sample of college students (n=84), where each row in the spreadsheet represents one student's response, measured in dollars. Can you reject the null hypothesis that the average college student spends $1,200 a year on textbooks at the one-percent (0.01) level of significance?
Show your work and explain using the critical value, p-value, and confidence interval approaches. Be sure to intuitively explain what your results imply. Use GRETL for this question.
6. Testing Proportions
Suppose someone claims that at least 90 percent of U.S. households have internet access. To test this claim, you randomly sample 100 U.S. households and find that 81 percent of your sample has internet access. What are the null and alternative hypotheses? What is the value of your test statistic? Can you reject the null hypothesis at the five-percent level of significance? Can you reject the null hypothesis at the one-percent level of significance? Show your work and explain.