Chapter 13 – Inference for Tables:

Chi-Square Procedures

Section 1

Test for Goodness of Fit

Note: Copy this document into your folder and rename it to YI_Ch13Example13_1.doc

Example 13.1: The graying of America

In recent years, the expression “the graying of America” has been used to refer to the belief that with better medicine and healthier lifestyles, people are living longer, and consequently a larger percentage of population is of retirement age. We want to investigate whether this perception is accurate. The distribution of the US population in 1980 is shown in Table 13.1. We want to determine if the distribution of the age groups in the US in 1996 has changed significantly from the 1980 distribution from the 1980 distribution. We will test the following hypothesis:

H0: the age group distribution of 1996 is the same as the 1980 age distribution.

Ha: the age group distribution of 1996 is different from the 1980 age distribution.

Table 13.1 US population by age group, 1980

Age group / Population (thousand) / Percent
0 to 24 / 93,777 / 41.39
25 to 44 / 62,716 / 27.68
45 to 64 / 44,503 / 19.64
65 and older / 25,550 / 11.28
Total / 226,546 / 100.00

H0: p0-24 = 0.4139, p25+44 = 0.2768, p56-64 = 0.1964, p65+ = 0.1128

Ha: at least one of the proportions differs from the stated values

Table 13.2 Sample results for 500 randomly selected

individuals in 1996

Age group / Count / Percent
0 to 24 / 177 / 35.4
25 to 44 / 158 / 31.6
45 to 64 / 101 / 20.2
65 and older / 64 / 12.8
500 / 100

Use Excel to create a segmented bar graph.

Table 13.3 Expected Counts

Age group / 1980
Population % / 1996
Expected Counts
0 to 24 / 41.39
25 to 44 / 27.68
45 to 64 / 19.64
65 and older / 11.28
100.00 / 500

Table 13.4 Calculating the goodness of fit

Age group / Observed / Expected /
0 to 24 / 177
25 to 44 / 158
45 to 64 / 101
65 and older / 64
=

Technology Toolbox

·  Clear L1 /list1, L2 /list2 and L3 /list3

·  Enter the observed counts in L1 /list1 and expected counts L2 /list2

·  Define L3 as (L1 – L2 )2 / L2 . [list3 as (list1-list2)2 /list2]

· 

·  Use the command sum(L3) to calculate in LIST/ MATH for TI 83

·  sum(list3) located in the CATALOG

·  Find the P-value using the cdf command (in the distribution (DISTR) menu on the TI-83 and in the CATALOG under Flash Apps on the TI-89)

1.  What can you say about this P-value?

2.  Compare it to the value for a critical P-value=0.05.

3.  Visualize the result by graphing the shaded area. Use Shade(,999,3). This command is located in the DISTR/DRAW menu on the TI 83 and in the CATALOG under Flash Apps on the TI-89. First, clear your screen (DRAW/ClrDraw) and choose the following WINDOW

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