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	497	1927	336	876	364

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: The probabilities are not all equal to 1/5.     H₀:  p₁ = p₂ = p₃ = p₄ = p₅ = 1/5H₀:  p₁ = 0.14, p₂ = 0.48, p₃ = 0.08, p₄ = 0.22, and p₅ = 0.08.

(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 H₀fail 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

The following table is obtained:

Categories	Observed	Expected	(fo-fe)2/fe
0.14	497	4000*0.14=560	(497-560)2/560 = 7.088
0.48	1927	4000*0.48=1920	(1927-1920)2/1920 = 0.026
0.08	336	4000*0.08=320	(336-320)2/320 = 0.8
0.22	876	4000*0.22=880	(876-880)2/880 = 0.018
0.08	364	4000*0.08=320	(364-320)2/320 = 6.05
Sum =	4000	4000	13.981

From the above calculated table

(a) Null hypothesis ( H0 ) :  p₁ = p₂ = p₃ = p₄ = p₅ = 1/5

(b) Test statistic :

χ2 = 13.981

(c) P-value :

With χ2 = 13.981 , df = 4

we get P value = 0.0074

( d ) conclusion :

P value < l.o.s

i.e., 0.0074 < 0.01

So reject H0

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