Question

Disadvantage groups, notably Blacks and Hispanics. Have had smaller high school graduation rates and so less access to college than Whites. Among those with college degrees is an educational success beyond college similarly affected? To address this question use the data below. sample of 30-year-old Americans with college degrees.

Highest Degree. Whites Black Hispanic Row Totals

College 5030 549 412 5991

Advanced 1324    117 99 1540

Columm totals 6354 668 511 7634 (grand)

a) state appropriate hypotheses

b) Find the degree of freedom

Compute the expected values for the entries for Hispanics. Compute the corresponding contributions to x^2. To save time here are the contributions of x² from the cells of white and blacks: 0.12, 0.7, 0.47, and 2.70.

d) conduct the appropriate test and give (an estimate of) p-value.

e) give an appropriate conclusion in statistical and everyday language.

Answer 1

Solution:

Here, we have to use chi square test for independence of two categorical variables.

a) state appropriate hypotheses

Null hypothesis: H₀: Among those with college degrees is an educational success beyond college not similarly affected.

Alternative hypothesis: H_a: Among those with college degrees is an educational success beyond college similarly affected.

We assume level of significance = α = 0.05

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

E = row total * column total / Grand total

b) Find the degree of freedom

We are given

Number of rows = r = 2

Number of columns = c = 3

Degrees of freedom = df = (r – 1)*(c – 1) = 1*2 = 2

α = 0.05

Critical value = 5.991465

(by using Chi square table or excel)

Compute the expected values for the entries for Hispanics.

Calculation tables for test statistic are given as below:

Observed Frequencies
	Column variable
Row variable	White	Black	Hispanic	Total
College	5030	549	412	5991
Advanced	1324	117	99	1540
Total	6354	666	511	7531

Expected Frequencies
	Column variable
Row variable	White	Black	Hispanic	Total
College	5054.683	529.8109	406.5066	5991
Advanced	1299.317	136.1891	104.4934	1540
Total	6354	666	511	7531

Calculations
(O - E)
-24.6825	19.18909	5.493427
24.68251	-19.1891	-5.49343

(O - E)^2/E
0.120527	0.695005	0.074237
0.468882	2.703748	0.2888

expected values for the entries for Hispanics = 0.07, 0.29

d) conduct the appropriate test and give (an estimate of) p-value.

Chi square = ∑[(O – E)^2/E] = 4.351199

P-value = 0.11354

(By using Chi square table or excel)

e) give an appropriate conclusion in statistical and everyday language.

P-value > α = 0.05

So, we do not reject the null hypothesis

There is insufficient evidence to conclude that among those with college degrees is an educational success beyond college similarly affected.