Lab 5 Objectives:

In this lab, we will use a few JAVA applets to help familiarize ourselves with sampling distributions, the CLT, consistency, and confidence intervals. Also, we'll use S-Plus to construct confidence intervals using data from exercises 12 and 13, Chapter 9.

Sampling Distributions and Central Limit Theorem Simulation

We'll look at two JAVA applets designed to illustrate Sampling Distributions and the Central Limit Theorem. The first one we've seen in class (at least there is a link to it in our notes). You can sample various sample sizes (n) repeatedly from the normal, binomial or uniform distributions. Read the description before running the applet

To use the applet, select a population distribution: First, select Population: Normal (mean=.5, sd=.200). Then select Sample size: n=5 for each sample from the population. Click on Show population and Show sampling distribution of mean. The distribution of the sample mean is shown in green while the original distribution from which you're sampling is in blue (in this case N(0.5, 0.04)). (The sampling distribution is calculated using well known methods which we may or may not cover in class, depending on the original distribution.) Click on Draw a Sample (repeat a couple of times - the red arrow indicates where the sample mean is). Now click on Show Obtained Means. Draw a single sample, or, to speed things up, Draw 100 Samples. The applet repeatedly calculates the sample average for each of the 100 samples of size n. Note that as the sample size, n, increases, the sample means tend to cluster more closely around the true population mean (0.5 in this case). This is because the sample mean from a random sample is "consistent" for the population mean. Note that the standard error reflects the sampling variability of the mean, depicted in the red histogram of means: as sample size, n, increases, we have more information, so the variability of the mean decreases. Also note how well the histogram in red follows the probability density in green. Try increasing the sample size n. Does the population distribution change? Does the sampling distribution of the mean change?

Now assume the underlying distribution is non-normal. Choose Population: Binomial p=.1 and Sample size: n=25

Experiment with the applet until you get a feel for the CLT and consistency.

The next JAVA applet is also designed to illustrate the Central Limit Theorem using sums of dots on dice. Try it!

Confidence Interval Simulations

Click on the link below to bring up a JAVA applet that simulates normal data and constructs (2-sided) confidence intervals. Each line in the plot represents a confidence interval for the mean based on simulated normal data with the population mean set to 0 and variance 1. Not all intervals will contain the population mean. Play with the simulator to see how the number of intervals that do not contain the mean changes with alpha. How do the lengths of the intervals change with alpha? If we could change the original population variance, how would an increase/decrease in variance change the CIs? How would a change in the sample size change the width of the CIs?

Bring up JAVA Confidence Interval Simulator

Exercises 12-13 Chapter 9

For exercise 12, download the dataset SERZINC or copy it from your CD-ROM and then read it into S-Plus

To find a confidence interval, use the menu: Statistics > Data Summaries > Summary Statistics.... Select zinc under the Variables list. Click on the Statistics tab. In this dialog box, click to place a check in the box for "Conf Limits for Mean". By default the level of the CI is 0.95; change as needed. Select any of the other outputs that you would like, then click OK. The results will appear in the Report Window, with LCL Mean corresponding to the lower limit of the confidence interval, and UCL being the upper confidence limit.

Follow the same approach for the questions in exercise 13 using the data set LOWBWT