IT 223: Data Analysis
Winter 2006

Assignment 5

Due Friday February 17 before 11:30pm by DL Web Submission

General chance problems

1. Four draws are going to be made at random with replacement from the box consisting of cards labeled as follows: 1, 2, 2, 3, 3. Find the chance that a card labeled "3" is drawn at least once.
2. The random variable X is constructed from drawing one card from the box of cards in the last problem. Determine the following:
1. Expected value (mean) of X.
2. Variance of X.
3. The expected value (mean) of X + 2.
4. The variance of X + 2.
5. The expected value (mean) of X + X.
6. The variance of X + X.
7. The standard deviation of X + X.
3. A coin is tossed some number of times. For each of the circumstances below, choose whether you would prefer 100 tosses or 1000 tosses.
   1. You win if heads is tossed at least 51% of the time.
   2. You win if heads is tossed at least 49% of the time.
   3. You win if heads is tossed between 49% and 51% of the time.
   4. You win if heads is tossed exactly 50% of the time.

Dice Rolling Simulations

DiceApplet is java applet that allows you to simulate rolling samples of 6-sided dice and reporting their sums for each roll. Through text fields, you may specify how many samples should be simulated and how many dice should be thrown for each sample. The output field lists the sums of the dice for each sample of dice that was simulated. You may copy this output and use it for the input to a SAS or Excel program.

Complete the following:

1. What is the expected value of rolling a 6-sided die?
2. What is the expected sum of rolling 10 6-sided dice?
3. What is the expected sum of rolling 100 6-sided dice?
4. Run the following simulations:
   • 10 samples of rolling 10 6-sided dice
   • 10 samples of rolling 100 6-sided dice
5. Answer the following from the above simulations:
   • On average for each sample of 10 dice, how far off was the actual sum of 10 dice from the expected sum of 10 dice (average absolute error)?
   • On average for each sample of 100 dice, how far off was the actual sum of 100 dice from the expected sum of 100 dice (average absolute error)?
   • For each sample of 10 dice, calculate the average roll within the sample. Then calculate how far off the average roll is from the expected value of rolling a 6-sided die.
   • For each sample of 100 dice, calculate the average roll within the sample. Then calculate how far off the average roll is from the expected value of rolling a 6-sided die.
   • Reviewing the differences between expected values and actual values, explain how these results are consistent with the law of large numbers.
6. What is the expected standard deviation of the sum of rolling 50 6-sided dice? (hint: find the variance of rolling one six-sided die, then the variance of 50 dice, then the standard deviation)
7. Run a simulation of 100 samples of 50 dice. Record their sums and create a frequency bar chart showing the sums of 100 samples. Insert this chart into your assignment. Consider the expected sum and the standard deviation from the last problem. Explain how your results are consistent with the Central Limit Theorem.

Submission

Submit a one-document report that responds to the numbered instructions.