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Statistics

11.2 Goodness-of-Fit Test

Statistics11.2 Goodness-of-Fit Test
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Table of contents
  1. Preface
  2. 1 Sampling and Data
    1. Introduction
    2. 1.1 Definitions of Statistics, Probability, and Key Terms
    3. 1.2 Data, Sampling, and Variation in Data and Sampling
    4. 1.3 Frequency, Frequency Tables, and Levels of Measurement
    5. 1.4 Experimental Design and Ethics
    6. 1.5 Data Collection Experiment
    7. 1.6 Sampling Experiment
    8. Key Terms
    9. Chapter Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  3. 2 Descriptive Statistics
    1. Introduction
    2. 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
    3. 2.2 Histograms, Frequency Polygons, and Time Series Graphs
    4. 2.3 Measures of the Location of the Data
    5. 2.4 Box Plots
    6. 2.5 Measures of the Center of the Data
    7. 2.6 Skewness and the Mean, Median, and Mode
    8. 2.7 Measures of the Spread of the Data
    9. 2.8 Descriptive Statistics
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  4. 3 Probability Topics
    1. Introduction
    2. 3.1 Terminology
    3. 3.2 Independent and Mutually Exclusive Events
    4. 3.3 Two Basic Rules of Probability
    5. 3.4 Contingency Tables
    6. 3.5 Tree and Venn Diagrams
    7. 3.6 Probability Topics
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Bringing It Together: Practice
    13. Homework
    14. Bringing It Together: Homework
    15. References
    16. Solutions
  5. 4 Discrete Random Variables
    1. Introduction
    2. 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
    3. 4.2 Mean or Expected Value and Standard Deviation
    4. 4.3 Binomial Distribution (Optional)
    5. 4.4 Geometric Distribution (Optional)
    6. 4.5 Hypergeometric Distribution (Optional)
    7. 4.6 Poisson Distribution (Optional)
    8. 4.7 Discrete Distribution (Playing Card Experiment)
    9. 4.8 Discrete Distribution (Lucky Dice Experiment)
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. References
    16. Solutions
  6. 5 Continuous Random Variables
    1. Introduction
    2. 5.1 Continuous Probability Functions
    3. 5.2 The Uniform Distribution
    4. 5.3 The Exponential Distribution (Optional)
    5. 5.4 Continuous Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  7. 6 The Normal Distribution
    1. Introduction
    2. 6.1 The Standard Normal Distribution
    3. 6.2 Using the Normal Distribution
    4. 6.3 Normal Distribution—Lap Times
    5. 6.4 Normal Distribution—Pinkie Length
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  8. 7 The Central Limit Theorem
    1. Introduction
    2. 7.1 The Central Limit Theorem for Sample Means (Averages)
    3. 7.2 The Central Limit Theorem for Sums (Optional)
    4. 7.3 Using the Central Limit Theorem
    5. 7.4 Central Limit Theorem (Pocket Change)
    6. 7.5 Central Limit Theorem (Cookie Recipes)
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  9. 8 Confidence Intervals
    1. Introduction
    2. 8.1 A Single Population Mean Using the Normal Distribution
    3. 8.2 A Single Population Mean Using the Student's t-Distribution
    4. 8.3 A Population Proportion
    5. 8.4 Confidence Interval (Home Costs)
    6. 8.5 Confidence Interval (Place of Birth)
    7. 8.6 Confidence Interval (Women's Heights)
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  10. 9 Hypothesis Testing with One Sample
    1. Introduction
    2. 9.1 Null and Alternative Hypotheses
    3. 9.2 Outcomes and the Type I and Type II Errors
    4. 9.3 Distribution Needed for Hypothesis Testing
    5. 9.4 Rare Events, the Sample, and the Decision and Conclusion
    6. 9.5 Additional Information and Full Hypothesis Test Examples
    7. 9.6 Hypothesis Testing of a Single Mean and Single Proportion
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  11. 10 Hypothesis Testing with Two Samples
    1. Introduction
    2. 10.1 Two Population Means with Unknown Standard Deviations
    3. 10.2 Two Population Means with Known Standard Deviations
    4. 10.3 Comparing Two Independent Population Proportions
    5. 10.4 Matched or Paired Samples (Optional)
    6. 10.5 Hypothesis Testing for Two Means and Two Proportions
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  12. 11 The Chi-Square Distribution
    1. Introduction
    2. 11.1 Facts About the Chi-Square Distribution
    3. 11.2 Goodness-of-Fit Test
    4. 11.3 Test of Independence
    5. 11.4 Test for Homogeneity
    6. 11.5 Comparison of the Chi-Square Tests
    7. 11.6 Test of a Single Variance
    8. 11.7 Lab 1: Chi-Square Goodness-of-Fit
    9. 11.8 Lab 2: Chi-Square Test of Independence
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  13. 12 Linear Regression and Correlation
    1. Introduction
    2. 12.1 Linear Equations
    3. 12.2 The Regression Equation
    4. 12.3 Testing the Significance of the Correlation Coefficient (Optional)
    5. 12.4 Prediction (Optional)
    6. 12.5 Outliers
    7. 12.6 Regression (Distance from School) (Optional)
    8. 12.7 Regression (Textbook Cost) (Optional)
    9. 12.8 Regression (Fuel Efficiency) (Optional)
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  14. 13 F Distribution and One-way Anova
    1. Introduction
    2. 13.1 One-Way ANOVA
    3. 13.2 The F Distribution and the F Ratio
    4. 13.3 Facts About the F Distribution
    5. 13.4 Test of Two Variances
    6. 13.5 Lab: One-Way ANOVA
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  15. A | Appendix A Review Exercises (Ch 3–13)
  16. B | Appendix B Practice Tests (1–4) and Final Exams
  17. C | Data Sets
  18. D | Group and Partner Projects
  19. E | Solution Sheets
  20. F | Mathematical Phrases, Symbols, and Formulas
  21. G | Notes for the TI-83, 83+, 84, 84+ Calculators
  22. H | Tables
  23. Index

In this type of hypothesis test, you determine whether the data fit a particular distribution. For example, you may suspect your unknown data fit a binomial distribution. You use a chi-square test, meaning the distribution for the hypothesis test is chi-square, to determine if there is a fit. The null and the alternative hypotheses for this test may be written in sentences or may be stated as equations or inequalities.

The test statistic for a goodness-of-fit test is:

Σ k (O−E) 2 E Σ k (O−E) 2 E

where

  • O = observed values (data),
  • E = expected values (from theory), and
  • k = the number of different data cells or categories.

The observed values are the data values, and the expected values are the values you would expect to get if the null hypothesis were true. There are n terms of the form (O−E) 2 E (O−E) 2 E .

The number of degrees of freedom is df = (number of categories – 1).

The goodness-of-fit test is almost always right-tailed. If the observed values and the corresponding expected values are not close to each other, then the test statistic can get very large and will be way out in the right tail of the chi-square curve.

Note

The expected value for each cell needs to be at least five for you to use this test.

Example 11.1

Absenteeism of college students from math classes is a major concern to math instructors because missing class appears to increase the drop rate. Suppose that a study was done to determine if the actual student absenteeism rate follows faculty perception. The faculty expected that a group of 100 students would miss class according to Table 11.1.

Number of Absences per Term Expected Number of Students
0–2 50
3–5 30
6–8 12
9–11 6
12+ 2
Table 11.1

A random survey across all mathematics courses was then done to determine the number of observed absences in a course. Table 11.2 displays the results of that survey.

Number of Absences per Term Actual Number of Students
0–2 35
3–5 40
6–8 20
9–11 1
12+ 4
Table 11.2


Determine the null and alternative hypotheses needed to conduct a goodness-of-fit test.

H0: Student absenteeism fits faculty perception.


The alternative hypothesis is the opposite of the null hypothesis.

Ha: Student absenteeism does not fit faculty perception.

Problem

a. Can you use the information as it appears in the charts to conduct the goodness-of-fit test?

Problem

b. What is the number of degrees of freedom (df)?

Try It 11.1

A factory manager needs to understand how many products are defective versus how many are produced. The number of expected defects is listed in Table 11.5.

Number Produced Number Defective
0–100 5
101–200 6
201–300 7
301–400 8
401–500 10
Table 11.5

A random sample was taken to determine the actual number of defects. Table 11.6 shows the results of the survey.

Number Produced Number Defective
0–100 5
101–200 7
201–300 8
301–400 9
401–500 11
Table 11.6

State the null and alternative hypotheses needed to conduct a goodness-of-fit test, and state the degrees of freedom.

Example 11.2

Problem

Employers want to know which days of the week employees are absent in a five-day work week. Most employers would like to believe that employees are absent equally during the week. Suppose a random sample of 60 managers were asked on which day of the week they had the highest number of employee absences. The results were distributed as in Table 11.7. For the population of employees, do the days for the highest number of absences occur with equal frequencies during a five-day work week? Test at a 5 percent significance level.

Monday Tuesday Wednesday Thursday Friday
Number of Absences 15 12 9 9 15
Table 11.7 Day of the Week Employees Were Most Absent

Try It 11.2

Teachers want to know which night each week their students are doing most of their homework. Most teachers think that students do homework equally throughout the week. Suppose a random sample of 56 students were asked on which night of the week they did the most homework. The results were distributed as in Table 11.8.

Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Number of Students 11 8 10 7 10 5 5
Table 11.8

From the population of students, do the nights for the highest number of students doing the majority of their homework occur with equal frequencies during a week? What type of hypothesis test should you use?

Example 11.3

One study indicates that the number of televisions that American families have is distributed (this is the given distribution for the American population) as in Table 11.9.

Number of Televisions Percent
0 10
1 16
2 55
3 11
4+ 8
Table 11.9

The table contains expected (E) percents.

A random sample of 600 families in the far western U.S. resulted in the data in Table 11.10.

Number of Televisions Frequency
0 66
1 119
2 340
3 60
4+ 15
Total = 600
Table 11.10

The table contains observed (O) frequency values.

Problem

At the 1 percent significance level, does it appear that the distribution number of televisions of far western U.S. families is different from the distribution for the American population as a whole?

Try It 11.3

The expected percentage of the number of pets students have in their homes is distributed (this is the given distribution for the student population of the United States) as in Table 11.12.

Number of Pets Percent
0 18
1 25
2 30
3 18
4+ 9
Table 11.12

A random sample of 1,000 students from the eastern United States resulted in the data in Table 11.13.

Number of Pets Frequency
0 210
1 240
2 320
3 140
4+ 90
Table 11.13

At the 1 percent significance level, does it appear that the distribution number of pets of students in the eastern United States is different from the distribution for the United States student population as a whole? What is the p-value?

Example 11.4

Problem

Suppose you flip two coins 100 times. The results are 20 HH, 27 HT, 30 TH, and 23 TT. Are the coins fair? Test at a 5 percent significance level.

Try It 11.4

Students in a social studies class hypothesize that the literacy rates around the world for every region are 82 percent. Table 11.14 shows the actual literacy rates around the world broken down by region. What are the test statistic and the degrees of freedom?

MDG Region Adult Literacy Rate (%)
Developed regions 99
Commonwealth of Independent States 99.5
Northern Africa 67.3
Sub-Saharan Africa 62.5
Latin America and the Caribbean 91
Eastern Asia 93.8
Southern Asia 61.9
Southeastern Asia 91.9
Western Asia 84.5
Oceania 66.4
Table 11.14
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