a) Analyze the following table that describes the relationship between the percent black and median income in Massachusetts census tracts in 2000.[1]
Table 1a. Median
Income by Percent Black, all MA census tracts, 2000.
|
Percent
Black Income |
Low % black (0-7%) |
Medium % black (7-50%) |
High % black (50%+) |
Total |
|
Lower Income $0-$40K |
11.3% (125) |
48.8% (101) |
68.9% (31) |
19% (257) |
|
Higher Income $40K+ |
88.7% (979) |
51.2% (106) |
31.1% (14) |
81% (1099) |
|
Total |
100% (1104) |
100% (207) |
100% (45) |
100% 1356 |
Table 1a examines differences in median income across census tracts in Massachusetts with different proportions of African-Americans. The table describes a strong association between the percent African-American in census tracts and the median income of tracts: as the proportion African-American increases, the median income of a tract decreases.
Among tracts where the African-American population was 50 percent or higher, just under one third of the tracts reported median incomes of $40,000 or more. In sharp contrast, among non-black tracts – that is, tracts that had 7 percent or fewer African-Americans – the vast bulk (88.7%) reported median incomes of $40,000 of more. This is a 57.6 percent difference in median income between low black and high black tracts in Massachusetts. The magnitude of this difference suggests that tracts with African-American majorities are much poorer on average than tracts with low proportions of African-Americans. Of course, these latter tracts are predominantly White…..
b) To compute the chi-square statistic, first compute the expected cell frequency for each cell of the crosstab. The expected frequencies are those one would expect given the univariate distributions (or marginals) of the variables in the table. They are calculated as
Expected Cell Frequency = (column total)*(row total)/N
For example, the computation of the expected
cell frequency for census tracts with lower income and low percent black would
be:
Table 1b. Example,
expected frequency for Table 1a.
|
Percent Black Income |
Low % black (0-7%) |
Medium % black (7-50%) |
High % black (50%+) |
Total |
|
Lower Income $0-$40K |
Expected (257*1104)/1356 = 209.2 |
|
|
257 |
|
Higher Income $40K+ |
|
|
|
|
|
Total |
1104 |
|
|
N=1356 |
c) Using the same procedure to compute all the expected frequencies yields the following table of expected frequencies:
Table 1c. Table of
expected frequencies for Table 1a.
|
|
|
Low % black (0-7%) |
Medium % black (7-50%) |
High % black (50%+) |
Total |
|
Lower Income $0-$40K |
Observed Expected |
125 209.2 |
101 39.2 |
31 8.5 |
257 257 |
|
Higher Income $40K+ |
Observed Expected |
979 894.8 |
106 167.8 |
14 36.5 |
1099 1099 |
|
Total |
|
1104 1104 |
207 207 |
45 45 |
1356 |
d)
The next step
is to subtract the expected cell frequency from the observed cell frequency for
each cell. This value gives the amount of the deviation or error for each cell.
Adding these to the preceding table yields the following.
Table 1d. Table of
observed-expected frequencies for Table 1a.
|
|
|
Low % black (0-7%) |
Medium % black (7-50%) |
High % black (50%+) |
Total |
|
Lower Income $0-$40K |
Observed Expected Obs-Exp |
125 209.2 -84.2 |
101 39.2 61.8 |
31 8.5 22.5 |
257 257 |
|
Higher Income $40K+ |
Observed Expected Obs-Exp |
979 894.8 84.2 |
106 167.8 -61.8 |
14 36.5 -22.5 |
1099 1099 |
|
Total |
|
1104 1104 |
207 207 |
45 45 |
1356 1356 |
Notice that the sum of the expected row total is the same as the sum for the observed row total; the same is true for the column totals. Note also that the sum of the observed – expected for both rows and columns equals zero.
e) Following this, the difference computed in the last step (i.e., the observed – expected) is squared, resulting in the following table:
Table 1e. Table of
(observed-expected)2 frequencies for Table 1a.
|
|
|
Low % black (0-7%) |
Medium % black (7-50%) |
High % black (50%+) |
Total |
|
Lower Income $0-$40K |
Observed Expected Obs-Exp (Obs-Exp)2 |
125 209.2 -84.2 7089.64 |
101 39.2 61.8 3819.24 |
31 8.5 22.5 506.25 |
257 257 |
|
Higher Income $40K+ |
Observed Expected Obs-Exp (Obs-Exp)2 |
979 894.8 84.2 7089.64 |
106 167.8 -61.8 3819.24 |
14 36.5 -22.5 506.25 |
1099 1099 |
|
Total |
|
1104 1104 |
207 207 |
45 45 |
1356 1356 |
f) Each of the squared differences is then divided by the expected cell frequency for each cell, resulting in the following table:
Table 1f. Table of
(observed-expected)2/expected frequencies for Table 1a.
|
|
|
Low % black (0-7%) |
Medium % black (7-50%) |
High % black (50%+) |
Total |
|
Lower Income $0-$40K |
Observed Expected Obs-Exp (Obs-Exp)2 (Obs-Exp)2/E |
125 209.2 -84.2 7089.64 33.9 |
101 39.2 61.8 3819.24 97.4 |
31 8.5 22.5 506.25 59.6 |
257 257 |
|
Higher Income $40K+ |
Observed Expected Obs-Exp (Obs-Exp)2 (Obs-Exp)2/E |
979 894.8 84.2 7089.64 7.9 |
106 167.8 -61.8 3819.24 22.8 |
14 36.5 -22.5 506.25 13.9 |
1099 1099 |
|
Total |
|
1104 1104 |
207 207 |
45 45 |
1356 1356 |
g) The chi-square statistic is computed by summing the last row of each cell in the preceding tables.
The computation for this example would result in the following:
Observed
chi-square = Sum (observed-expected)2
Expected
= 33.9+97.4+59.6+7.9+22.8+13.9
= 235.5[2]