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

i	1	2	3	4	5
	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

i	1	2	3	4	5
	No	High School	Associate's	Bachelor's	Graduate or
	Diploma	Diploma	Degree	Degree	Professional Degree
Count	493	1927	336	876	368

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 in terms of the probabilities of the outcomes?

H₀: The distribution in 2013 is different from that in 2005.

H₀:  p₁ = p₂ = p₃ = p₄ = p₅ = 1/5    

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

H₀: The probabilities are not all equal to 1/5.

(b) What is the value of the test statistic? Round to 3 decimal places unless your software automatically rounds to 2 decimal places.

χ²

=

(c) Use software to get the P-value of the test statistic. Round to 4 decimal places unless your software automatically rounds to 3 decimal places.
P-value =

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

reject H₀

fail to reject H₀    

(e) 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) Correct answer: H₀:  p₁ = 0.14, p₂ = 0.48, p₃ = 0.08, p₄ = 0.22, and p₅ = 0.08.

b)

	Observed		Expected
Type of Patient	Freq (Oi)	Probability	Freq. (Ei)	(Oi-Ei)^2 /Ei
No Diploma	493	0.14	560	8.0161
High school	1927	0.48	1920	0.0255
Associate's Degree	336	0.08	320	0.8000
Bachelor's Degree	876	0.22	880	0.0182
Professional Degree	368	0.08	320	7.2000
Total:	4000	1.00	4000	16.0598

Num Categories: 5
Degrees of freedom: 4

Test Statistic, X^2: 16.0598

c) P-Value: 0.0029

d) Since P-value < alpha 0.01 so we reject H0

Correct answer: Option (A) reject H₀

e) Correct answer: Option(B) The data suggests that the distribution of 2013 education attainment levels is different from the distribution in 2005.