Question

12.53 AIDS Cases. Refer to Exercise 12.47. For AIDS cases in the United States in 2011, solve the following problems:

Region	White	Black	Other	Total
Northeast	1,100	2,493		5,177
Northwest	1,137		504	3221
South	2,761	7,848		12,867
West		764	1766	4,230
Total			6052	25,435

a. Find and interpret the conditional distributions of region by race.

b. Find and interpret the marginal distribution of region.

c. Are the variables "region" and "race" associated? Explain your answer.

d. What percentage of AIDS cases were in the South?

e. What percentage of AIDS cases among whites were in the South?

f. Without doing further calculations, respond true or false to the distributions of race by region are not identical.

g. Find and interpret the marginal distribution of race and the conditional distributions of race by region.

Short answer

(a)

Region	White	Black	Other	Total
Northeast	1,100	2,493		5,177
Northwest	1,137		504	3221
South	2,761	7,848		12,867
West		764	1766	4,230
Total			6052	25,435

(b)

Region	White	Black	Other	Marginal distribution
Northeast	1,100	2,493	1,524	0.201
Northwest	1,137	1,580	504	0.127
South	2,761	7,848	2,258	0.506
West	1,700	764	1766	0.166
Total	6,698	12,685	6052	1

(c)Yes, the differences between "region" and "race" are associated because the conditional distribution of gender within levels is not the same.

(d) 50.6%

(e)41.2%

(f)

The given statement is true because the variables are associated .

(g)

Region	White	Black	Other
Northeast	0.164	0.197	0.252
Northwest	0.170	0.125	0.083
South	0.412	7,848	0.373
West	0.254	764	0.292
Marginal distribution	0.263	0.499	0.238

Step-by-step solution

Step 1. Given

The given contingency table is

Region	White	Black	Other	Total
Northeast	1,100	2,493		5,177
Northwest	1,137		504	3221
South	2,761	7,848		12,867
West		764	1766	4,230
Total			6052	25,435

Step 2. Part (a) Determine the conditional distribution of region

Region	White	Black	Other
Northeast	0.164	0.197	0.252
Northwest	0.170	0.125	0.083
South	0.412	7,848	0.373
West	0.254	764	0.292
Total	1	1	1

Step 3. Part(b) Determine the marginal distribution

Region	White	Black	Other	Marginal distribution
Northeast	1,100	2,493	1,524	0.201
Northwest	1,137	1,580	504	0.127
South	2,761	7,848	2,258	0.506
West	1,700	764	1766	0.166
Total	6,698	12,685	6052	1

Step 4. Part (c) Determine whether  the variables "region"  and "race" are associated or not

Yes, the differences between "region" and "race" are associated because the conditional distribution of gender within levels is not the same.

Step 5. Part(d) Determine the % of AIDS cases in south

50.6%

Step 6. Part(e) Determine the % of AIDS cases  among whites in south 

41.2%

Step 7. Part (f) Determine whether the statement is true or false

The given statement is true because the variables are associated .

Step 8. Part (g) Find and interpret the marginal distribution of race and the conditional distributions of race by region. 

Region	White	Black	Other
Northeast	0.164	0.197	0.252
Northwest	0.170	0.125	0.083
South	0.412	7,848	0.373
West	0.254	764	0.292
Marginal distribution	0.263	0.499	0.238