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Lab 1: Chi-Square Goodness-of-Fit

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

Summary: This module provides a lab on Chi-Square Distribution as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Class Time:

Names:

Student Learning Outcome:

  • The student will evaluate data collected to determine if they fit either the uniform or exponential distributions.

Collect the Data

Note:

You may need to combine two categories so that each cell has an expected value of at least 5.

Go to your local supermarket. Ask 30 people as they leave for the total amount on their grocery receipts. (Or, ask 3 cashiers for the last 10 amounts. Be sure to include the express lane, if it is open.)

  1. 1. Record the values.
    Table 1
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
    __________ __________ __________ __________ __________
  2. 2. Construct a histogram of the data. Make 5 - 6 intervals. Sketch the graph using a ruler and pencil. Scale the axes.
    Figure 1
    Blank graph with relative frequency on vertical
  3. 3. Calculate the following:
    • a. x¯=x¯= size 12{ {overline {x}} } {}
    • b. s=s= size 12{s} {}
    • c. s2=s2= size 12{s rSup { size 8{2} } } {}

Uniform Distribution

Test to see if grocery receipts follow the uniform distribution.

  1. 1. Using your lowest and highest values, XX ~ U_______,_______U_______,_______ size 12{X "~" U left ("_______, _______" right )} {}
  2. 2. Divide the distribution above into fifths.
  3. 3. Calculate the following:
    • a. Lowest value =
    • b. 20th percentile =
    • c. 40th percentile =
    • d. 60th percentile =
    • e. 80th percentile =
    • f. Highest value =
  4. 4. For each fifth, count the observed number of receipts and record it. Then determine the expected number of receipts and record that.
    Table 2
    Fifth Observed Expected
    1st    
    2nd    
    3rd    
    4th    
    5th    
  5. 5. HoHo size 12{H rSub { size 8{o} } } {}:
  6. 6. HaHa size 12{H rSub { size 8{a} } } {}:
  7. 7. What distribution should you use for a hypothesis test?
  8. 8. Why did you choose this distribution?
  9. 9. Calculate the test statistic.
  10. 10. Find the p-value.
  11. 11. Sketch a graph of the situation. Label and scale the x-axis. Shade the area corresponding to the p-value.
    Figure 2
    Blank graph with vertical and horizontal axes.
  12. 12. State your decision.
  13. 13. State your conclusion in a complete sentence.

Exponential Distribution

Test to see if grocery receipts follow the exponential distribution with decay parameter 1 x¯ 1 x .

  1. 1. Using 1x¯1x¯ size 12{ { {1} over { {overline {x}} } } } {} as the decay parameter, XX ~ Exp_______Exp_______ size 12{X "~" ital "Exp" left ("_______" right )} {}.
  2. 2. Calculate the following:
    • a. Lowest value =
    • b. First quartile =
    • c. 37th percentile =
    • d. Median =
    • e. 63rd percentile =
    • f. 3rd quartile =
    • g. Highest value =
  3. 3. For each cell, count the observed number of receipts and record it. Then determine the expected number of receipts and record that.
    Table 3
    Cell Observed Expected
    1st    
    2nd    
    3rd    
    4th    
    5th    
    6th    
  4. 4. HoHo size 12{H rSub { size 8{o} } } {}
  5. 5. HaHa size 12{H rSub { size 8{a} } } {}
  6. 6. What distribution should you use for a hypothesis test?
  7. 7. Why did you choose this distribution?
  8. 8. Calculate the test statistic.
  9. 9. Find the p-value.
  10. 10. Sketch a graph of the situation. Label and scale the x-axis. Shade the area corresponding to the p-value.
    Figure 3
    Blank graph with vertical and horizontal axes.
  11. 11. State your decision.
  12. 12. State your conclusion in a complete sentence.

Discussion Questions

  1. Did your data fit either distribution? If so, which?
  2. In general, do you think it’s likely that data could fit more than one distribution? In complete sentences, explain why or why not.

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