Data Analysis

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ST314 Data Analysis #1

Topics:

• Discrete Random Variables, Probability Mass Functions, Expectations. • Lessons Covered: 1- 7 • Textbook Chapter (Optional) : 2 and 3

Grading:

• Points are listed next to each question and should total 20 points overall. • To encourage better understanding, you may have your data analysis pre-graded with feedback. After

viewing the feedback, you may make changes and resubmit your analysis for final grading. • View feedback by clicking Grades and your assignment. Feedback will be written on your document. • Late assignments will not be graded. Lowest two data analysis grades will be dropped.

Deadlines:

• Optional pre-grading: Friday September 28th - Returned with feedback by Sunday. • Final Submission: Monday October 1st - Returned with feedback by following Saturday.

Instructions:

• Download or view questions below. • Clearly label and type answers, without question prompts, in word, google docs, PDF, or other word

processing software. • Insert diagrams or plots as a picture in an appropriate location. • Math Formulas need to be typed with Math Type, LateX, or clearly using key board symbols such as +, -. *,

/, sqrt() and ^ • Submit assignment. Verify the correct document has been uploaded. If not, resubmit. You can submit up to

three times.

Allowances:

• Any resources listed or posted in our class. • You are encouraged to discuss the problems with other students, the instructor and TAs, however, all work

must be your own words. Duplicate wording will be considered plagiarism. • Outside Resources need to be cited. Websites such as chegg, coursehero, koofers are discouraged, but if

used need to be cited and used within the boundaries of academic honesty.

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Part 1. (6 points) For each random variable, state whether the random variable should be modeled with a Binomial distribution or a Poisson distribution. Explain your reasoning. State the parameter values that describe the distribution and give the probability mass function. Random Variable 1. A quality measurement for cabinet manufacturers is whether a drawer slides open and shut easily. Historically, 2% of drawers fail the easy slide test. A manufacturer samples 10 drawers from a batch. Assuming the chance of failure is independent between drawers, what type of distribution could be used to model the number of failed drawers from the sample of 10? Random Variable 2. The warranty for a particular system on a new car is 2 years. During which there is no limit to the number of warranty claims per car. Historically, the average number of claims per car during the period is 0.8 claims. What type of distribution could be used to model the number of warranty claims per car? Part 2: (8 points) Wheel of Fortune is a popular game show on Television. Contestants spin a wheel and try to guess a correct letter from a word puzzle. If they guess correctly, they earn the dollar amount from the wheel. If they spin “bankrupt” or “lose a turn” they get nothing and can’t play. To the right is an example of the wheel. Watch this video to see an example of someone spinning the wheel. https://www.youtube.com/watch?v=_Pv33JWBdY8 The outcome of a spin on the wheel is a discrete random variable. Consider X the dollar amount spun on the wheel, where Bankrupt and Lose a Turn = $0, and Free Play = $500. There are 24 wedges on the wheel.

The following table provides the probability mass function for X. Round values to 3 decimal places.

a. (1 point) What is the most likely dollar amount? b. (1 point) How likely is it to spin Lose a turn or Bankrupt? c. (3 points) What is the average dollar amount? Show work! d. (2 points) Suppose a contestant spins the wheel three times, how likely is it they spin $0 each time? Show

work! e. (2 points) Suppose a contestant spins the wheel three times, how likely is it they spin $0 at least one time?

Show work!

x $0 $300 $350 $400 $450 $500 $550 $600 $700 $800 $900 $5,000 Count 2 5 1 2 1 3 1 3 1 2 2 1

P(x) 0.083 0.208 0.042 0.083 0.042 0.125 0.042 0.125 0.042 0.083 0.083 0.042

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Part 3: (6 points) The PMF in Part 2 is based on probability theory aka “math”. Do these probabilities stand up when a contestant actually spins the wheel? Go back to the Data Analysis #1 instructions page. Download the R script titled: Wheel_of_Fortune_Spin_Script.R , open the file it will automatically open in R. You need R software on the computer to open the script window. Follow the instructions in the code then answer the following: a. (1 point) What value did you spin? b. (1 point) What is the average of the 1000 simulated spins? How different is this from part 2c? c. (1 point) Paste the probability mass function and the plot of the probability mass function from R. d. (1 point) How different are the simulated probabilities to the theoretical probabilities in part 2? e. (1 point) Based on the plot is the most likely outcome the same as it is in part 2a? f. (1 point) In general, what action will make the simulated values more like the theoretical ones?