Question

Changes in Education Attainment: According to the U.S. Census Bureau, the distribution of Highest Education Attainment in U.S. adults aged 25 - 34 in the year 2005 is given in the table below.

Census: Highest Education Attainment - 2005

	No	High School	Associate's	Bachelor's	Graduate or
	Diploma	Diploma	Degree	Degree	Professional Degree
Percent	14%	48%	8%	22%	8%

In a survey of 4000 adults aged 25 - 34 in the year 2013, the counts for these levels of educational attainment are given in the table below.

Survey (n = 4000): Highest Education Attainment - 2013

	No	High School	Associate's	Bachelor's	Graduate or
	Diploma	Diploma	Degree	Degree	Professional Degree
Count	485	1922	336	876	381

The Test: Test whether or not the distribution of education attainment has changed from 2005 to 2013. Conduct this test at the 0.01 significance level.(a) What is the null hypothesis for this test?

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 1/5 H₀: The probabilities are not all equal to 1/5.     H₀: p₁ = 0.14, p₂ = 0.48, p₃ = 0.08, p₄ = 0.22, and p₅ = 0.08. H₀: The distribution in 2013 is different from that in 2005.

(b) The table below is used to calculate the test statistic. Complete the missing cells.
Round your answers to the same number of decimal places as other entries for that column.

	Highest	Observed	Assumed	Expected
i	Education	Frequency (Oi)	Probability (pi)	Frequency Ei	(Oi − Ei)2 Ei
1	No Diploma	485	0.14	560	2
2	Diploma	1922	0.48	3	0.002
3	Associate's	336	4	320	0.800
4	Bachelor's	5	0.22	880	0.018
5	Grad or Prof	381	0.08	320	11.628
Σ		n = 4000			χ2 = 6

(c) What is the value for the degrees of freedom? 7
(d) What is the critical value of

χ²

? Use the answer found in the

χ²

-table or round to 3 decimal places.
t_α = 8

(e) What is the conclusion regarding the null hypothesis?

reject H₀ fail to reject H₀    

(f) Choose the appropriate concluding statement.

We have proven that the distribution of 2013 education attainment levels is the same as the distribution in 2005. The data suggests that the distribution of 2013 education attainment levels is different from the distribution in 2005.      There is not enough data to suggest that the distribution of 2013 education attainment levels is different from the distribution in 2005.

Answer 1

a) Null and Alternative hypothesis:

H0: p1 = 0.14, p2 = 0.48, p3 = 0.08, p4 = 0.22, and p5 = 0.08.

Category	Observed Frequency (O)	Proportion, p	Expected Frequency (E)	(O-E)²/E
No diploma	485	0.14	4000 * 0.14 = 560	(485 - 560)²/560 = 10.0446
High school diploma	1922	0.48	4000 * 0.48 = 1920	(1922 - 1920)²/1920 = 0.0021
Associate's Degree	336	0.08	4000 * 0.08 = 320	(336 - 320)²/320 = 0.8
Bachelor's Degree	876	0.22	4000 * 0.22 = 880	(876 - 880)²/880 = 0.0182
Graduate Degree	381	0.08	4000 * 0.08 = 320	(381 - 320)²/320 = 11.6281
Total	4000	1	4000	χ² = 22.4930

c) Degree of freedom, df = n-1 = 4

d) Critical value:

χ²α = CHISQ.INV.RT(0.01, 4) = 13.277

e) As χ² = 22.493 > 13.277, so we reject the null hypothesis.

f) Conclusion:

The data suggests that the distribution of 2013 education attainment levels is different from the distribution in 2005.