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Continuous Random Variables: Lab I

Module by: Barbara Illowsky, Ph.D., Susan Dean. E-mail the authors

Summary: In this lab exercise, students will compare and contrast empirical data from a random number generator with the Uniform Distribution. Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.

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Class Time:

Names:

Student Learning Outcomes:

  • The student will compare and contrast empirical data from a random number generator with the Uniform Distribution.

Collect the Data

Use a random number generator to generate 50 values between 0 and 1 (inclusive). List them below. Round the numbers to 4 decimal places or set the calculator MODE to 4 places.

  1. Complete the table:
    Table 1
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
  2. Calculate the following:
    • a. x¯=x¯= size 12{ {overline {x}} ={}} {}
    • b. s=s= size 12{s={}} {}
    • c. 40th percentile =
    • d. 3rd quartile =
    • e. Median =

Organizing the Data

  1. Construct a histogram of the empirical data. Make 8 bars.
    Figure 1
    Blank graph with relative frequency on the vertical axis and X on the horizontal axis.
  2. Construct a histogram of the empirical data. Make 5 bars.
    Figure 2
    Blank graph with relative frequency on the vertical axis and X on the horizontal axis.

Describe the Data

  1. Describe the shape of each graph. Use 2 – 3 complete sentences. (Keep it simple. Does the graph go straight across, does it have a V shape, does it have a hump in the middle or at either end, etc.? One way to help you determine a shape, is to roughly draw a smooth curve through the top of the bars.)
  2. Describe how changing the number of bars might change the shape.

Theoretical Distribution

  1. In words, X X =
  2. The theoretical distribution of XX is XX ~ U(0,1)U(0,1). Use it for this part.
  3. In theory, based upon the distribution in the section titled "Organizing the Data",
    • a. μ = μ=
    • b. σ = σ=
    • c. 40th percentile =
    • d. 3rd quartile =
    • e. median = __________
  4. Are the empirical values (the data) in the section titled "Collect the Data" close to the corresponding theoretical values above? Why or why not?

Plot the Data

  1. Construct a box plot of the data. Be sure to use a ruler to scale accurately and draw straight edges.
  2. Do you notice any potential outliers? If so, which values are they? Either way, numerically justify your answer. (Recall that any DATA are less than Q1 – 1.5*IQR or more than Q3 + 1.5*IQR are potential outliers. IQR means interquartile range.)

Comparing the Data

  1. For each part below, use a complete sentence to comment on how the value obtained from the data compares to the theoretical value you expected from the distribution in the section titled "Theoretical Data".
    • a. minimum value:
    • b. first quartile:
    • c. median:
    • d. third quartile:
    • e. maximum value:
    • f. width of IQR:
    • g. overall shape:
  2. Based on your comments in the section titled "Collect the Data", how does the box plot fit or not fit what you would expect of the distribution in the section titled "Theoretical Distribution"?

Discussion Question

  1. Suppose that the number of values generated was 500, not 50. How would that affect what you would expect the empirical data to be and the shape of its graph to look like?

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