Chance Variability--- Answers to Sample Questions
1. A box contains 200 red marbles and 800 green marbles. Nine hundred marbles are drawn at random with replacement . Which of the following statements are true and why?
A. About 720 of the draws are going to be green give or take a dozen are so.
True.Expected value = (number of draws) x ( average of box) = 900 x (800/1000) = 720
B. Exactly 20 percent of the draws are going to be red.
False. According to the law of averages, it is likely to be close to 20%
C. There is about a 50 percent chance that between 172 and 188 draws are going to be red.
True. The expected value for the sum is 180 and the standard error for the sum is square root of 900 x 0.2 x 0.8 which is 12 so the interval is -.67 to +.67 standard units. (To convert to standard units subtract the expected value the divide by the standard error.) The area under the normal curve over this interval is about 50%
2. A programmer is working on a new program, KERRICH 2.0, to simulate tossing a coin. As a preliminary test of the output of this program, she plans to set up the program to do one million tosses one thousand times.
A. Show that there is about a _95%___percent chance of getting between 499020 and 500980 heads in one million tosses.
Build a box model.
What goes in the box? 50 percent 1's and 50 percent 0's.
What is the average of the box? 0.5 the SD of the box? 0.5
How many draws from the box? 1,000,000.
What is the expected value for the sum of the draws from the box? 500,000 the standard error? 500
Now use the normal table and fill in the blank! 980/500 = 1.96 and the interval 499020 to 500980 heads is -1.96 to 1.96 standard units. The area under the normal curve over this interval is about 95%.
B. In 1000 runs of 1 million simulated tosses KERRICH 2.0 should get between 499020 and 500980 heads _95.0_____percent of the time give or take _0.7_____percent or so.
Build a box model.
What goes in the box? 95 1's and 5 0's.
What is the average of the box? 0.95 the SD of the box? 0.2179
How many draws from the box? 1000
What is the expected value for the percentage of ones drawn from the box? 95 the standard error? 0.69