MATH 324 University of California Riverside Cumulative Probability Worksheet

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Math 324 Probability and Statistics with Computing
Project 3 Confidence Intervals
Objective: Explore a confidence interval estimation through simulation and random sampling.
You may use MS Excel, Google Sheets, or any other statistical software of your choice. If you
use R or Python, you will submit the source code and the output.
Introduction: Here’s the game: there are two boxes with four balls each. To play, you put on a
blindfold and then pick one ball at random from each box.
There are two red, one blue and one green ball in the box on the left. There are two blue, one red
and one green ball in the box on the right.
Payouts:
•
•
•
•
$0 if the balls are different colors,
$0.50 if they are both red,
$1 if they are both blue, and
$16 if they are both green.
It costs $1 to play the game.
Let 𝑋𝑋 be the overall amount of money you end up with (won or lost) after playing the game. This
includes the cost of the game. For instance, if on a given game you win $16 but paid $1 to play the
game, you walk away with $15. Thus, in this case 𝑋𝑋 = $15. On the other hand, if you did not win
($0) but paid $1 to play, then 𝑋𝑋 = −$1.
Project Instructions:
1. State the probability distribution of the variable 𝑋𝑋.
2. Find the mean (𝜇𝜇) and the standard deviation (𝜎𝜎) of 𝑋𝑋.
3. Randomly sample data from this distribution. Generate 1000 samples of size 𝑛𝑛 = 50.
4. Summarize the data generated (50000 values) into a relative frequency table to confirm the
data follow the theoretical distribution in step 1.
5. Compute a sample mean from the data in each sample. There should be 1000 sample
means.
6. Make a histogram of the sample means. You may use the bins of your choice. (According
to the CLT, the distribution of the 𝑋𝑋� variable is approximately normal. Does the graph look
bell-shaped?)
7. Choose a 95% confidence level and use 𝜎𝜎 from step 2 to compute the associated margin of
error.
8. For each sample, compute a 95% 𝑧𝑧-confidence interval. There should be 1000 confidence
intervals.
9. What percentage of these confidence intervals contain the true population mean you found
in step 2?
Some MS Excel function you may need to complete this project:
RAND, VLOOKUP, SUMPRODUCT, COUNTIF, COUNTIFS, AVERAGE,
NORM.S.INV, CONFIDENCE.NORM
Be sure to organize your work in such a way that it looks presentable and is easy to follow!
Math 324 Probability and Statistics with Computing
Project 3 Confidence Intervals
Objective: Explore a confidence interval estimation through simulation and random sampling.
You may use MS Excel, Google Sheets, or any other statistical software of your choice. If you
use R or Python, you will submit the source code and the output.
Introduction: Here’s the game: there are two boxes with four balls each. To play, you put on a
blindfold and then pick one ball at random from each box.
There are two red, one blue and one green ball in the box on the left. There are two blue, one red
and one green ball in the box on the right.
Payouts:
•
•
•
•
$0 if the balls are different colors,
$0.50 if they are both red,
$1 if they are both blue, and
$16 if they are both green.
It costs $1 to play the game.
Let 𝑋𝑋 be the overall amount of money you end up with (won or lost) after playing the game. This
includes the cost of the game. For instance, if on a given game you win $16 but paid $1 to play the
game, you walk away with $15. Thus, in this case 𝑋𝑋 = $15. On the other hand, if you did not win
($0) but paid $1 to play, then 𝑋𝑋 = −$1.
Project Instructions:
1. State the probability distribution of the variable 𝑋𝑋.
2. Find the mean (𝜇𝜇) and the standard deviation (𝜎𝜎) of 𝑋𝑋.
3. Randomly sample data from this distribution. Generate 1000 samples of size 𝑛𝑛 = 50.
4. Summarize the data generated (50000 values) into a relative frequency table to confirm the
data follow the theoretical distribution in step 1.
5. Compute a sample mean from the data in each sample. There should be 1000 sample
means.
6. Make a histogram of the sample means. You may use the bins of your choice. (According
to the CLT, the distribution of the 𝑋𝑋� variable is approximately normal. Does the graph look
bell-shaped?)
7. Choose a 95% confidence level and use 𝜎𝜎 from step 2 to compute the associated margin of
error.
8. For each sample, compute a 95% 𝑧𝑧-confidence interval. There should be 1000 confidence
intervals.
9. What percentage of these confidence intervals contain the true population mean you found
in step 2?
Some MS Excel function you may need to complete this project:
RAND, VLOOKUP, SUMPRODUCT, COUNTIF, COUNTIFS, AVERAGE,
NORM.S.INV, CONFIDENCE.NORM
Be sure to organize your work in such a way that it looks presentable and is easy to follow!

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