1. Chapter 12: Inference about a Population Lecture 6b
Instructor: Naveen Abedin

2. THE CHI-SQUARE DISTRIBUTION
Definition
The chi-square distribution has only one parameter called the degrees of freedom. The shape of a chi-squared distribution curve is skewed to the right for small df and becomes symmetric for large df. The entire chi-square distribution curve lies to the right of the vertical axis. The chi-square distribution assumes nonnegative values only, and these are denoted by the symbol χ2 (read as “chi-square”).
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3. Three chi-square distribution curves.
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4. Example 1
Find the value of χ² for 7 degrees of freedom and an area of .10 in the right tail of the chi-square distribution curve.
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5. χ2 for df = 7 and .10 Area in the Right Tail
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6. Figure 1
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7. Example 2
Find the value of χ² for 12 degrees of freedom and an area of .05 in the left tail of the chi-square distribution curve.
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8. Area in the right tail = 1 – Area in the left tail = 1 – .05 = .95
Example 2: Solution
Area in the right tail = 1 – Area in the left tail = 1 – .05 = .95
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9. Table 11.2 χ2 for df = 12 and .95 Area in the Right Tail
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10. Figure 11.3
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11. Inferences about Population Variance
Population mean measures the population’s centrality, while population variance measures the variability that exists in the population.
If we are interested in investigating the variability in population, we have to investigate
As in most real world cases, the population variance is unknown. This means we need to estimate this value from sample variance.
Variance is associated with the words variability, risk, consistency
Variance is a measure of risk, and it is favorable to keep this value small
Variance is also important to check for consistency, e.g. consistency of product quality can be ensured through consistency in the production process. So if the production process has a low variance then we can ensure quality is consistent.

12. Inferences about Population Variance (cont.)
sample variance is considered as the best estimator of population variance because the sample variance is an unbiased and consistent estimator of population variance.
The chi-squared statistic can be used to make estimations about the population variance, at n – 1 degrees of freedom

13. Inferences about Population Variance (cont.)
Confidence interval of chi-squared statistic:

14. Example 2

15. Inferences about Population Variance (cont.)

16. Example 3
You have taken a random sample of 25 boxes of Brand A cereal, and hypothesized that the standard deviation has changed from the previously published level of 15 grams. The standard deviation of your sample is Test your theory at 5% level of significance.

17. Example 4
In a July 23, 2009, Harris Interactive Poll, 1015 advertisers were asked about their opinions of Twitter. The percentage distribution of their responses is shown in the following table.
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18. Example 4
Assume that these percentage hold true for the 2009 population of advertisers. Recently 800 randomly selected advertisers were asked the same question. The following table lists the number of advertisers in this sample who gave each response.
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19. Example 4
Test at the 2.5% level of significance whether the current distribution of opinions is different from that for 2009.
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20. Example 4: Solution Step 1:
H0 : The current percentage distribution of opinions is the same as for 2009.
H1 : The current percentage distribution of opinions is different from that for 2009.
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21. We use the chi-square distribution to make this test.
Example 4: Solution
Step 2:
There are 4 categories
5 days on opinion
Multinomial experiment
We use the chi-square distribution to make this test.
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22. Area in the right tail = α = .025 k = number of categories = 4
Example 4: Solution
Step 3:
Area in the right tail = α = .025
k = number of categories = 4
df = k – 1 = 4 – 1 = 3
The critical value of χ2 = 9.348
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24. Calculating the Value of the Test Statistic
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25. Example 4: Solution Step 4:
All the required calculations to find the value of the test statistic χ2 are shown in Table 11.4.
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26. Hence, we fail to reject the null hypothesis
Example 4: Solution
Step 5:
The value of the test statistic χ2 = is smaller than the critical value of χ2 = 9.348
It falls in the nonrejection region
Hence, we fail to reject the null hypothesis
We state that the current percentage distribution of opinions is the same as for 2009.
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27. Example 5
A researcher wanted to study the relationship between gender and owning cell phones. She took a sample of 2000 adults and obtained the information given in the following table.
At the 5% level of significance, can you conclude that gender and owning cell phones are related for all adults?
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28. H0: Gender and owning a cell phone are not related
Example 5: Solution
Step 1:
H0: Gender and owning a cell phone are not related
H1: Gender and owning a cell phone are related
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29. We are performing a test of independence
Example 5: Solution
Step 2:
We are performing a test of independence
We use the chi-square distribution
Step 3:
α = .05.
df = (R – 1)(C – 1) = (2 – 1)(2 – 1) = 1
The critical value of χ2 = 3.841
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30. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

31. Observed and Expected Frequencies
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32. Example 5: Solution Step 4:
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33. The value of the test statistic χ2 = 21.445
Example 5: Solution
Step 5:
The value of the test statistic χ2 =
It is larger than the critical value of χ2 = 3.841
It falls in the rejection region
Hence, we reject the null hypothesis
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34. Example 6
Consider the data on income distributions for households in California and Wisconsin given in Table Using the 2.5% significance level, test the null hypothesis that the distribution of households with regard to income levels is similar (homogeneous) for the two states.
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35. Table 11.10 Observed Frequencies
Example 6
Table Observed Frequencies
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36. Example 6: Solution Step 1:
H0: The proportions of households that belong to different income groups are the same in both states
H1: The proportions of households that belong to different income groups are not the same in both states
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37. Step 2: We use the chi-square distribution to make a homogeneity test.
Example 6: Solution
Step 2: We use the chi-square distribution to make a homogeneity test.
Step 3:
α = .025
df = (R – 1)(C – 1) = (3 – 1)(2 – 1) = 2
The critical value of χ2 = 7.378
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38. Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

39. Observed and Expected Frequencies
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40. Example 6: Solution Step 4:
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41. The value of the test statistic χ2 = 4.339
Example 6: Solution
Step 5:
The value of the test statistic χ2 = 4.339
It is less than the critical value of χ2
It falls in the nonrejection region
Hence, we fail to reject the null hypothesis
We state that the distribution of households with regard to income appears to be similar (homogeneous) in California and Wisconsin.
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