Statistics 2501 (001)
Assignment #1: Sept. 12, 2003
Due in class: Sept. 24, 2003

• All problem numbers are from the textbook Statistics for Business and Economics, 8th Edition.

1. This problem will get you to use the world of www to perform a small simulation study to get a feel for the concept of the sampling distribution of $\bar{x}$ and the central limit theorem. (I'm sorry to report that no M & M's are involved). What you will do is run a program that will draw repeated samples from a normal distribution and observe the behaviour of $\bar{x}$. This will be a multi-part exercise.
   1. Once logged in to the computer, start up Internet Explorer or Netscape and go to the following site:
      

      www.ruf.rice.edu/~lane/rvls.html
      

      Once at this site, click on
      

      Simulations/Demonstrations - Sampling Distribution Simulation
      

      then the Begin box on the left-hand side of the screen.

   2. A histogram appears at the top of the page, depicting a normal distribution. What are you told the mean ($\mu$) and standard deviation ($\sigma$, written as sd on the screen) are?

   3. You are going to get the computer to draw samples from the distribution shown in (a), and calculate the mean of each of the samples. You will first do this by drawing 5 samples from this distribution. Each sample will contain $n = 20$ observations. NOTE: The computer screen calls this $N$.

      To do this, go to the third plot area on the screen, entitled Distribution of Means. Change the value of $N$ to $N = 20$.

      Then, go by the second plotting area and click the box 5 Samples. This will choose 5 samples, each with 20 values, and find the mean of each sample ($\bar{x}$).

   4. A histogram of these $\bar{x}$ should appear. Describe its shape.

   5. What are the mean and and standard deviation of the $\bar{x}$ values? How do they compare with what the Central Limit Theorem says the theoretical mean ($\mu$) and standard deviation ( $\sigma/\sqrt{20}$) of the sampling distribution will be?

   6. Next, click on the box that says 1,000 Samples. This will draw 1000 samples from your normal distribution, and calculate the corresponding 1000 $\bar{x}$ values.

   7. Describe the shape of the new histogram that is produced.

   8. What are the mean and standard deviation of these $\bar{x}$ values? Are they ``closer'' to what the Central Limit Theorem says should happen?

2. Refer to problem #8.36, p. 359, but change $s = 7.1$ in the problem to $\sigma = 6.5$. Now answer the following questions:
   1. 8.36(a).

   2. Calculate the p-value for the test used in (a), and draw an appropriate conclusion.

   3. Construct a 90% confidence interval for the true mean age of MSNBC news viewers.

3. Problem #10.71(a, b), p. 523-524. Also, use your least squares line to predict the time it would take to fill an order of 100 cases. Do this problem by hand.

4. Refer to the data in #10.20, p. 471-472, and complete the following:
   1. Plot the data, either by hand or using Minitab, and comment on the relationship between household buying income and retail sales at eating and drinking places.

   2. #10.20(a), using Minitab.

   3. Graph the least squares line by hand on the scatterplot you created in part (a).

   4. Interpret $\hat{\beta}_{0}$ and $\hat{\beta}_{1}$ in the context of this problem.

   5. Predict the retail sales at eating and drinking places for a state with average household buying income of $41,400.