Lab 5 (Week of 2/23/98)

1) Law of large numbers. The law of large numbers says that as sample size grows, the sample mean tends toward the population mean; technically, the likelihood of drawing a sample with a mean more than a given fixed distance from the population mean diminishes as sample size increases. Hence, the more data you collect, the more likely the sample mean (a typical estimate of the population mean) will be close to the population mean. The following Java applet simulates a data set drawn from one of three types of population. As the data is "collected" its histogram and a vertical line representing the sample mean are updated (both plotted in blue) above a background plot of the parent population with a vertical bar at its mean (both plotted in red). Click here to start the JAVA applet (programmed by Craig Merrill at BYU). Run the simulation for each of the three parent populations (Normal with mean 0 and variance 1, exponential with mean 1 and variance 1 and uniform with mean 1/2 and variance 1/12). For each run answer the following questions:
a) What is the mean of the resulting sample?
b) What is the sample size?
c) What is the variance?
d) What is the standard error of the mean?
e) How many standard errors above/below the true mean is the sample mean?

2) Central Limit Theorem. The central limit theorem states that the distribution of the sample average becomes more normal as sample size increases. The following Java applet simulates throws of fair die (1 or more at a time). For each throw it calculates the total of the exposed faces and updates the bar in a histogram associated with this total (to get the average of exposed faces, just divide the labels on the histogram's horizontal axis by the number of die rolled each time). In this setting, the central limit theorem says that the more die that go into calculating each total, the more nearly normal the distribution of resulting totals will be. Click here to start the applet (written by R. Todd Ogden, Dept. of Statistics, Univ. of South Carolina), then answer the following questions:
a) Repeatedly roll one die (at first one at time, then increase the number of rolls per click), what is the shape of the distribution?
b) Repeatedly roll two die (at first one at time, then increase the number of rolls per click), what is the shape of the distribution?
c) Repeatedly roll five die (at first one at time, then increase the number of rolls per click), what is the shape of the distribution?

3) Normal Approximation to the Binomial. We mentioned in class that one use of the normal distribution is to approximate binomial probabilities. Suppose a candidate has 67% support among voters. Suppose we draw a very simple random sample of 50 voters. How good an approximation to the sample proportion of voters supporting the candidate is the normal distribution? Use this binomial approximation JAVA applet (courtesy of David Lane at Rice University) to get a graphical answer. If the candidate's support was 99% would the approximation be better or worse? Use this normal probabilities JAVA applet (courtesy of Balasubramanian Narasimhan of Stanford University) to answer the following questions:
a) Assuming 67% support, what is the probability that the sample proportion will be greater than 0.5?
b) Assuming 67% support, what is the probability that the sample proportion will be between 0.5 and 0.7?

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last updated 21 February 1998