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Ivo Dinov
UCLA Statistics, Neurology, LONI, UCLA Statistics

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STAT 13 (1a, 1b, 1c)	*Introduction to Statistical Methods for the Life and Health Science*
Department of Statistics	Instructor: Ivo Dinov
Homework 5	Due Date: Friday, Nov. 19, 2004

Please, submit your homework right after lecture on the due date. See the HW submission rules. On the front page include the following header.

(HW_5_1) The casino game of Tai Sai is played by betting on various possible combinations of results when three dice are rolled. One possibility is to place a bet on a single number. In this case, the more times this number appears on the three dice, the greater the winnings. Here's an example of how it works: Consider a gambler who wishes to bet on the number six. For each dollar bet, the player wins twelve times his/her stake if three sixes are rolled on the three dice, two times his/her stake if two sixes are rolled on the three dice or one times his/her stake if only one six is rolled on the three dice. If no sixes are rolled, then the stake is lost.

Let X be the return from a single bet with a $5 stake. X has the following probability function:

x	-$5	$5	$10	$60
pr(X=x)	0.581	0.346	0.064	0.009

A player will make a profit overall if their mean return from placing a series of bets is positive. Let X_bar be the mean return from 100 separate bets with a $5 stake.

What are the values of E[X_bar] and SE[X_bar]?

What is the approximate distribution of X_bar? What theorem was needed to decide this?

Calculate the probability of, on average, making a positive return from placing one hundred separate bets with $5 stakes.

Repeat the calculations above using 1,000, 5,000 and 10,000 bets.

Create a scatter plot of probability that the mean return from playing Tai Sai is positive (on y-axis) versus number of bets (on x-axis). Comment on what the plot appears to show.

(HW_5_2) Successful TV advertising depends on being able to get viewers to recall the specific brand being advertised. Two groups of randomly selected students are shown the TV ads for 10 products. One groups saw the TV ads with some sexual content, the other saw the TV ads rated for general audience. One week after seeing the ads each students was asked to name the specific brands which they saw on TV. The mean and SD of the number of correct answers for each group are given in the table below.

Calculate SE(X₁_bar - X₂_bar), the standard error of the differences of the sample means for the two groups.
Calculate the two standard error interval for the difference of the two population means µ₁ - µ_2.
Write a one paragraph report summarizing your conclusions from the analysis of these data.

Sexual Content Ads	n₁=40	X₁_bar=6.9	S₁=1.87
General Audience Ads	n₂=58	X₂_bar=5.1	S₂=1.51

(HW_5_3) The fuel consumption, in liters per 100 kilometers, of a (small) car of a particular model has a mean of 6.21 and a standard deviation of 1.3 (both in litters). A random sample of these cars is taken.

Calculate the expected value and the standard deviation of the sample average if:

one observation is taken.
four observations are taken.
sixteen observations are taken.

In what way do your answers above differ? Why?
A larger model of the same car consumes on average 1.3 times the amount of fuel (of the smaller model) less 1.3 liters (per 100 km). If three small and five large cars of this make are randomly selected what would be the combined fuel consumption of all these 8 cars together per 100 km? How variable would this combined fuel consumption be from one sample (of 3 + 5) of cars to another?

(HW_5_4) You need to provide at least two examples graphically illustrating the effects of the central limit theorem (CLT). Include pictures (e.g., for small and large sample-sizes), descriptions and interpretations of these demos/experiments. For example, you may want to discuss the effects of the dice probabilities and the number of dice on the distribution of the average (of all dice) in the DiceExperiment in SOCR. You may also want to experiment with manually drawing a bimodal distribution and investigating the effects of the sample-size and the number of samples taken on the Normality of the sampling distribution of the sample average using the SamplingDistribution applet. Another possibility is to study the BinomialCoinExperiment in SOCR and discuss the effects of the probability of observing a head (success) and the number of coins on the distribution of the variable representing the number of successes in a number of such experiments. Yet another option is to use STATA to generate n-samples of size k-observations each and plot the histogram of the n sample averages. One should observe that this histogram has symmetric, bell-shapped and unimodal properties (at least approximately) - does the approximation improve as the sample-size increases?

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