Changes in Education Attainment: According to the U.S. Census Bureau, the distri

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

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

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: 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.

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

2

=  

(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.

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%

Explanation / Answer

Solution

This is a case of Chi-square test for goodness of fit.

Part (a)

Null hypothesis: H0: The distribution in 2013 is the same as that in 2005. i.e.,

H0:  p1 = 0.14, p2 = 0.48, p3 = 0.08, p4 = 0.22, and p5 = 0.08. ANSWER option (4)

Part (b)

Test statistic

2 = [i = 1,5]{(Oi - Ei)2/Ei }where Oi = observed frequency in the ith category in 2013 and Ei = expected frequency in the ith category as seen in 2005, which is obtained by multiplying the given frequency percentage by n (= 4000 given).

Excel calculations are tabulated below:

i

1

2

3

4

5

No

High School

Associate's

Bachelor's

Graduate or

Diploma

Diploma

Degree

Degree

Professional Degree

Total

Count 2013 (Oi)

527

1922

336

876

339

4000

Percent 2005

14

48

8

22

8

100

Ei = 4000 x %

560

1920

320

880

320

4000

(Oi - Ei)^2/Ei

1.944643

0.00208

0.8

0.01818

1.128125

3.89303

So, the calculated value of 2 = 3.893 ANSWER [Additional input: check: sum of oi must equal sum of Ei]

Part (c)

Under H0 , 2 ~ 2k – 1 where k = number of classes (categories) = 5.

So, p-value = P(24 > 3.893) = 0.4206 ANSWER

Part (d)

Conclusion: Since p-value > 0.01 (given level of significance), H0 is accepted. ANSWER

Part (e)

Concluding statement:

There is not enough data to suggest that the distribution of 2013 education attainment levels is different from the distribution in 2005. ANSWER

i

1

2

3

4

5

No

High School

Associate's

Bachelor's

Graduate or

Diploma

Diploma

Degree

Degree

Professional Degree

Total

Count 2013 (Oi)

527

1922

336

876

339

4000

Percent 2005

14

48

8

22

8

100

Ei = 4000 x %

560

1920

320

880

320

4000

(Oi - Ei)^2/Ei

1.944643

0.00208

0.8

0.01818

1.128125

3.89303

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