Question

Changes in Education Attainment: USE SOFTWARE - 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	535	1927	336	886	316

The Test: Test whether or not the distribution of education attainment has changed from 2005 to 2013. Conduct this test at the 0.05 significance level.

(a) What is the null hypothesis for this test?

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 1/5

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

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) 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	535	0.14	560
2	Diploma	1927	0.48		0.026
3	Associate's	336		320	0.800
4	Bachelor's		0.22	880	0.041
5	Grad or Prof	316	0.08	320	0.050
Σ		n = 4000			χ2 =

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

(d) What is the critical value of χ²? Use the answer found in the χ²-table or round to 3 decimal places.
t_α =  

(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) H0: p1 = 0.14. p2 = 0.48, p 3 = 0.08, p4 = 0.22 and p5 = 0.08

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(b) The complete table is as below. The missing cells are marked in yellow

	Observed	Expected %	Expected	(O-E)2	(O-E)2/E
No Diploma	535	0.14	560	625	1.116
Diploma	1927	0.48	1920	49	0.026
Associates	336	0.08	320	256	0.800
Bachelors	886	0.22	880	36	0.041
Grad or Prof	316	0.08	320	16	0.050
Total	4000.00	1.00	4000.00	982.00	2.033

= 2.033

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(c) Degrees of freedom = n - 1 = 5 - 1 = 4

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(d) critical = 9.488

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(e) Since = 2.033 is < critical = 9.488, we fail to reject H0.

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(f) Option 3: There is not enough data to suggest that the distribution of 213 education attainment levels is different from the distribution in 2005.

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