Question

Support of Background Checks by Political Party: In April of 2013, the U.S. Senate did not pass a bill to expand background checks to all gun sales despite popular approval of the idea. Gallup conducted a poll on this issue with the question: Would you vote for or against a law to require background checks for all gun sales?. The results by political affiliation are summarized in the contingency table below.

Observed Frequencies: O_i's

	Republican	Independent	Democrat	Totals
For Checks	284	75	310	669
Against Checks	76	13	42	131
Totals	360	88	352	800

The Test: Test for a dependent relationship between party affiliation and opinion on expanded background checks. Conduct this test at the 0.05 significance level.

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

H₀: Party affiliation and opinion on expanded background checks are dependent variables.

H₀: Party affiliation and opinion on expanded background checks are independent variables.      

(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 opinion on expanded background checks and party affiliation are independent.

The evidence suggests that there is a dependent relationship between party affiliation and opinion on expanded background checks.  

There is not enough evidence to conclude that party affiliation and opinion on expanded background checks are dependent.

Answer 1

Solution:

Part a

Here, we have to use chi square test for independence of two categorical variables.

Null hypothesis: H₀: Party affiliation and opinion on expanded background checks are independent variables.   

Alternative hypothesis: H_a: Party affiliation and opinion on expanded background checks are not independent variables.

We are given level of significance = α = 0.05

Part b

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

E = row total * column total / Grand total

We are given

Number of rows = r = 2

Number of columns = c = 3

Degrees of freedom = df = (r – 1)*(c – 1) = 1*2 = 2

α = 0.05

Critical value = 5.991465

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Column variable
Row variable	Rep	Ind	Demo	Total
For checks	284	75	310	669
Against checks	76	13	42	131
Total	360	88	352	800

Expected Frequencies
	Column variable
Row variable	Rep	Ind	Demo	Total
For checks	301.05	73.59	294.36	669
Against checks	58.95	14.41	57.64	131
Total	360	88	352	800

Calculations
(O - E)
-17.05	1.41	15.64
17.05	-1.41	-15.64

(O - E)^2/E
0.965629	0.027016	0.830988
4.93134	0.137967	4.243747

Chi square = ∑[(O – E)^2/E] = 11.13669

Chi square test statistic = 11.13669

Part c

P-value = 0.004

(By using Chi square table or excel)

Part d

P-value < α = 0.05

So, we reject the null hypothesis

Answer: Reject H₀

Part e

The evidence suggests that there is a dependent relationship between party affiliation and opinion on expanded background checks.