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UCLA Statistics, Neurology, LONI, Math/PIC

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Statistics 157 Spring 2011	*Probability and Statistics Data Modeling and Analysis using SOCR*
Department of Statistics	Instructor: Ivo Dinov
Homework 4	Due Date: Friday, May 20, 2011

Please, submit your homework before class on the due date. Complete at least 6 problems for full credit. See the HW submission rules. On the front page include the following header.

(HW_4_1) The pmf for X = {the number of major defects of a randomly selected appliance of a certain type} is given in the table below:

x	0	1	2	3	4
p(x)	0.08	0.15	0.45	0.27	0.05

) = E(

) − µ

(HW_4_2) An appliance dealer sells three different models of upright freezers having 13.5 ft³, 15.9 ft³, and 19.1 ft³ (cubic feet) of storage space, respectively. Let X = {the amount of freezer storage space purchased by the next customer}. Suppose that X has the following pmf.

x (ft³) 13.5 15.9 19.1

p(x) 0.2 0.5 0.3

(a) Compute the first three moments of the random variable X, E(X), E(X²), E(X³), and V(X).
(b) If the price of a freezer having capacity X ft³ is 25X − 8.5, what is the expected price paid by the next customer to buy a freezer?
(c) What is the variance of the price 25X − 8.5 paid by the next customer? Interpret this value!
(d) Suppose that although the rated capacity of a freezer is X, the actual capacity is h(X) = X − 0.01X², slightly under the advertised space. What is the expected actual capacity of the freezer purchased by the next customer?

(HW_4_3) Use the Binomial Coin Experiment to compare the theoretical and empirically observed mean and standard deviation of this SOCR Experiment. Run 10 times the experiment with number-of-coins n=14 and P(Head)=p=0.3. How different are the population (theoretical) and the sample (observed) mean and standard deviation? How many runs of the 14-coin-toss experiment you need to do to get both of these differences within 10^-2? If you do another 10 runs are these differences guaranteed to stay within this limit (do not refresh the experiment, do an extra 10 runs)? Look at the SOCR Law of Large Numbers Activity and explain in your own words!

(HW_4_4) Suppose you are randomly clipping coupons out of published materials. Each week before you go to the market you keep clipping coupons until you have 20 distinct coupons (there are no duplicates). Suppose there are 50 distinct vendors that advertise and print coupons in your local media. Using the SOCR Coupon Collector Experiment do the following:

Select the proper values for the parameters m (population-size) and k (number of distinct objects desired). Look at the corresponding distribution for the random variable W, which represents the total number of coupons clipped by shopping time. Without running any experiments, just look at the blue model distribution graph and the discrete model distribution table for W below, to estimate the odds that you will need to clip at least 35 coupons to get 20 distinct ones, as desired.
Now, run 1,000 experiments in auto-mode. In what proportion of these you had to clip at least 35 coupons to get 20 distinct ones, as desired?
What is the discrepancy between the theoretical probability and the empirically observed odds of having to clip at most 24 coupons, in the above 1,000 experiments?

(HW_4_5) For each of the following experiments, report your simulation and clearly indicate the smallest number of experiments (n) that provides the desired approximation (the values of n will be different each time you run the experiment, but you should keep running the experiment until you get an approximation within the specified limits):

Use the Uniform E-Estimate Experiment (see SOCR Law of Large Numbers Activity) to generate an approximation to the natural number ( e ) with error within 0.01.
Use the Buffon’s Needle Experiment (see the Buffon's Needle Activity) to estimate the number $π$ with error within 0.1.
Suppose a call service center expects to get 35 calls a minute for questions regarding each of 23 different vendors that rely on this call center for handling their calls. Use the Normal approximation to Poisson distribution to compute the probability that in a 1-minute interval they receive less than 710 calls in total? See this demonstration as example of Normal distribution approximation to Poisson.

(HW_4_6) Show empirically the following approximations: Normal to Binomial, Binomial to Hypergeometric, Normal to Poisson and Poisson to Binomial. Use SOCR Distributions and SOCR Experiments. Can you find empirical situations when these approximations fail? Please provide specifics, snapshots, try to explain why, etc.

(HW_4_7) What is a practical application of the Central Limit Theorem? Choose one answer. Explain.
- It gives you the message that you should pick as large of a sample as you can afford.
- It relates to the normal distribution and the concept of Z and percentile.
- It shows you the meaning of standard deviation over repeated samples.
- It allows you to make inference about the population mean and percentage.

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x (ft³)	13.5	15.9	19.1
p(x)	0.2	0.5	0.3

Instructor: Ivo Dinov