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Practice 1: Goodness-of-Fit Test

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

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

Student Learning Outcomes

  • The student will conduct a goodness-of-fit test.

Given

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as follows: (Source: HIV/AIDS Epidemiology Santa Clara County, Santa Clara County Public Health Department, May 2011)

Table 1
Ethnicity Number of Cases
White 2229
Hispanic 1157
Black/African-American 457
Asian, Pacific Islander 232
  Total = 4075

The percentage of each ethnic group in Santa Clara County is as follows:

Table 2
Ethnicity Percentage of total county population Number expected (round to 2 decimal places)
White 42.9% 1748.18
Hispanic 26.7%  
Black/African-American 2.6%  
Asian, Pacific Islander 27.8%  
  Total = 100%  

Expected Results

If the ethnicity of AIDS victims followed the ethnicity of the total county population, fill in the expected number of cases per ethnic group.

Goodness-of-Fit Test

Perform a goodness-of-fit test to determine whether the make-up of AIDS cases follows the ethnicity of the general population of Santa Clara County.

Exercise 1

H o : H o : size 12{H rSub { size 8{O} } :μ rSub { size 8{W} } =μ rSub { size 8{ ital "NW"} } } {}

Exercise 2

H a : H a : size 12{H rSub { size 8{a} } :μ rSub { size 8{W} } <> μ rSub { size 8{ ital "NW"} } } {}

Exercise 3

Is this a right-tailed, left-tailed, or two-tailed test?

Exercise 4

degrees of freedom =

Solution

degrees of freedom = 3

Exercise 5

Chi2Chi2 size 12{ ital "Chi" rSup { size 8{2} } } {} test statistic =

Solution

2016.14

Exercise 6

p-value =

Solution

Rounded to 4 decimal places, the p-value is 0.0000.

Exercise 7

Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade in the region corresponding to the p-value.

9.png

Let α = 0.05 α=0.05

Decision:

Reason for the Decision:

Conclusion (write out in complete sentences):

Discussion Question

Exercise 8

Does it appear that the pattern of AIDS cases in Santa Clara County corresponds to the distribution of ethnic groups in this county? Why or why not?

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