Question

Consider the following contingency table, which you will use to conduct a χ2 (Chi2) test.

• Sociological tidbit: participation in pre-K (pre-school) has been shown to correlate with a LOT of later-in-life positive outcomes - from elementary school performance, to high school performance, to higher employment rates, to higher incomes, and lower numbers of arrests over the life course. Many sociologists of education agree that investing in pre-K is a huge step towards reducing inequality in society [Source (Links to an external site.)]

Here is fake data for the purposes of this problem - let's say this is data from a random sample of 30 year olds, asking about school achievements and pre-K participation.

	Public Pre-K (Headstart, etc.)	Private Pre-K	No Pre-K	TOTAL
Did Not Graduate from High School	25	50	25	100
Graduated from High School	75	150	25	250
TOTAL	100	200	50	350

A. What is your alternative hypothesis? (4 pts)
B. What are the degrees of freedom for this problem? (2 pts)
C. What is your alpha level for this problem? (2 pts)
D. How many respondents were observed to have attended public or private pre-K and graduated from high school? (Round to whole number if necessary). (4 pts)
E. How many respondents were expected to have attended public or private pre-K and did not graduate from high school? (Round to whole number if necessary). (4 pts)
F. What is your calculated χ2 (Chi2) statistic?  (Round to two decimal places). (4 pts)
G. Using the Bognar Chi2 App [Link (Links to an external site.)], what is the returned alpha level given your χ2 (Chi2) statistic and your df? (2 pts)
H. What decision do you make about the alternative hypothesis? Explain why. (2 pts)
I. Report your final results in a full sentence (as exemplified on your worksheets from class). (4 pts)
J. Based on the results of this test alone, can we say that pre-K participation CAUSES improvements in high school graduation rates? Why or why not? Explain.

Answer 1

	Pre-K	No Pre-K	TOTAL
Did Not Graduate from High School	75	25	100
Graduated from High School	225	25	250
TOTAL	300	50	350

A. What is your alternative hypothesis? (4 pts)

H1: There is association between pre - K participation and school achievements.
B. What are the degrees of freedom for this problem? (2 pts)

df = (2-1)*(2-1) = 1
C. What is your alpha level for this problem? (2 pts)

0.05 (lets say)
D. How many respondents were observed to have attended public or private pre-K and graduated from high school? (Round to whole number if necessary). (4 pts)

250
E. How many respondents were expected to have attended public or private pre-K and did not graduate from high school? (Round to whole number if necessary). (4 pts)

F. What is your calculated χ2 (Chi2) statistic?  (Round to two decimal places). (4 pts)

The expected values are computed in terms of row and column totals. In fact, the formula is where Ri​ corresponds to the total sum of elements in row i, Cj​ corresponds to the total sum of elements in column j, and T is the grand total. The table below shows the calculations to obtain the table with expected values:

Expected Values	Pre K	No Pre K	Total
Not graduate	\ = 85.714350300×100​=85.714	\ = 14.28635050×100​=14.286	100
Graduate	\= 214.286350300×250​=214.286	= 35.71435050×250​=35.714	250
Total	300	50	350

Based on the observed and expected values, the squared distances can be computed according to the following formula: (E - O)^2/E(E−O)2/E. The table with squared distances is shown below:

Squared Distances	Pre K	No Pre K
Not graduate	\= 1.33985.714(75−85.714)2​=1.339	\ = 8.03614.286(25−14.286)2​=8.036
Graduate	= 0.536214.286(225−214.286)2​=0.536	= 3.21435.714(25−35.714)2​=3.214

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H0​: The two variables are independent

Ha​: The two variables are dependent

This corresponds to a Chi-Square test of independence.

Rejection Region

Based on the information provided, the significance level is \α=0.05 , the number of degrees of freedom is df=(2−1)×(2−1)=1, so then the rejection region for this test is R={χ2:χ2>3.841}.

Test Statistics

The Chi-Squared statistic is computed as follows:

Decision about the null hypothesis

Since it is observed that χ2=13.125>χc2​=3.841, it is then concluded that the null hypothesis is rejected.

Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the two variables are dependent, at the 0.05 significance level.

The corresponding p-value for the test is p=Pr(χ12​>13.125)=0.0003

G. Using the Bognar Chi2 App [Link (Links to an external site.)], what is the returned alpha level given your χ2 (Chi2) statistic and your df? (2 pts) I have no idea for this
H. What decision do you make about the alternative hypothesis? Explain why. (2 pts)

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the two variables are dependent, at the 0.05 significance level.
I. Report your final results in a full sentence (as exemplified on your worksheets from class). (4 pts)

We can say that participation in Pre-K influence success at stages of life by improving rates at school or college.
J. Based on the results of this test alone, can we say that pre-K participation CAUSES improvements in high school graduation rates? Why or why not? Explain.

Yes, as explained above.

Please rate my answer and comment for doubt