To do these exercises, go to www.macmillanhighered.com/fapp10e.

Question 8.117

4. The central limit theorem is the basis for the confidence intervals that have been discussed in this chapter and in Chapter 7 (page 321). Next, you will use the Central Limit Theorem applet to generate individual data values from two different continuous probability models: the uniform probability model and the exponential probability model. You will find data from these distributions don’t look very normal, and then you will take samples of size 30 and generate means from the samples.

  1. Go to the Central Limit Theorem applet. Choose “Uniform” for the distribution. Set the sample size to 1, so that you can see the results of individual data values drawn from this distribution. (The “Show normal curve” should be unchecked.) Click the “Generate samples” button. Do data from the uniform distribution appear to have the characteristic normal shape?
  2. Now change the sample size to 10. Instead of generating individual outcomes from a uniform distribution, the applet will draw many samples of size 10 and then make a histogram of the sample means, . Click the “Generate Samples” button. Do these data appear to be from a normal distribution? Check the box for “Show normal curve.”
  3. This time, choose “Exponential” for the distribution. Set the sample size to 1 as you did in part (a). Click the “Generate samples” button. Describe the shape of exponential data.
  4. Now, continue with the exponential distribution but change the sample size to 10. Instead of generating individual outcomes from an exponential distribution, the applet will draw many samples of size 10 and then make a histogram of the sample means, . Click the “Generate samples” button. Do these data appear to be from a normal distribution? Check the box for “Show normal curve.”
  5. Repeat part (d), but this time change the sample size to 30.
  6. Summarize the patterns you have observed in parts (a) through (e). How do these patterns relate to the central limit theorem?