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Discrete Random Variables: Lab I (edited: Teegarden)

Module by: Mary Teegarden Based on: Discrete Random Variables: Lab I by Dr. Barbara Illowsky, Susan Dean

Summary: This module allows students to explore concepts related to discrete random variables through the use of a simple playing card experiment. Students will compare empirical data to a theoretical distribution to determine if the experiment fist a discrete distribution. This lab involves the concept of long-term probabilities. Labs changed to incorporate mini-tabs.

Class Time:

Name:

Student Learning Outcomes:

  • The student will compare empirical data and a theoretical distribution to determine if everyday experiment fits a discrete distribution.
  • The student will demonstrate an understanding of long-term probabilities.

Supplies:

  • One full deck of playing cards

Procedure

The experiment procedure is to pick one card from a deck of shuffled cards.

  1. The theorectical probability of picking a diamond from a deck is:
  2. Shuffle a deck of cards.
  3. Pick one card from it.
  4. Record whether it was a diamon or not a diamond.
  5. Put the card back and reshuffle.
  6. Do this a total of 10 times
  7. Record the number of diamonds picked.
  8. Let X=X= number of diamonds. Theoretically, XX ~

Organize the Data

  1. Using Minitab, simulate this experiment for a total of 50 times. Use Calc -> Random data -> Binomial. Use Stats -> Tables -> Tally to summarize the data. Record the results in the chart below. Then calculate the relative frequency.
    X Frequency Relative Frequency
    0 __________ __________
    1 __________ __________
    2 __________ __________
    3 __________ __________
    4 __________ __________
    5 __________ __________
    6 __________ __________
    7 __________ __________
    8 __________ __________
    9 __________ __________
    10 __________ __________
  2. Calculate the following:
    • a. x¯ x =
    • b. ss =
  3. Construct a bar chart of the empirical data.
    Figure 1
    Blank graph with relative frequency on the vertical axis and number of diamonds on the horizontal axis.

Theorectical Distribution

  1. Using Minitab, build the theoretical PDF chart for X based on the distribution in the section above.
    X X size 12{X} {} P X = x P X = x size 12{P left (X=x right )} {}
    0  
    1  
    2  
    3  
    4  
    5  
    6  
    7  
    8  
    9  
    10  
  2. Calculate the following:
    • a. μ=μ= size 12{μ={}} {}________________________
    • b. σ=σ= size 12{σ={}} {}________________________
  3. Constuct a bar chart of the theoretical distribution.
    Figure 2
    Blank graph with relative frequency on the vertical axis and number of diamonds on the horizontal axis.

Using the Data

Calculate the following, rounding to 4 decimal places:

Note:

RF = relative frequency

Use the Theoretical probability table generated by Minitab here:

  • P ( X = 3 = P(X=3=
  • P ( 1 < X < 4 ) = P(1<X<4)=
  • P ( X ≥ 8 = P(X≥8=

Use the data from the simulation here:

  • RF ( X = 3 ) = RF(X=3)=
  • RF ( 1 < X < 4 ) = RF(1<X<4)=
  • RF ( X ≥ 8 ) = RF(X≥8)=

Discussion Questions

  1. Knowing that data vary, describe three similarities between the graphs and distributions of the theoretical and empirical distributions. Use complete sentences. (Note: These answers may vary and still be correct.)
  2. Describe the three most significant differences between the graphs or distributions of the theoretical and empirical distributions. (Note: These answers may vary and still be correct.)
  3. Does it appear that the data fit the distribution in Part I? In 1 - 3 complete sentences, explain why or why not.
  4. Suppose that the experiment had been repeated 500 times. Which chart (from Part II or Part III) would you expect to change? Why? Why wouldn’t the other chart change? How might the chart change?

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