Question

Use the following information for questions 12-16. A marketing research firm recently conducted a survey to find out about individuals' Internet buying behavior. 100 people responded. The firm now wants to know if the response to the question of whether they have purchased anything over the Internet is independent of gender. Below is the 2x2 Chi-squared contingency table data. Men - 27 have purchased (cell 1,1), 5 have not purchased (cell 1,2), total is 32. Women - 36 have purchased (cell 2,1), 32 have not purchased(cell 2,2), total is 68. Total who have purcahsed is 63 and total who have not purchased is 37. Total number of candidates = 100.

What is the expected value for cell 1,2?

What is the observed value for cell 2,1?

What is the calculated value of the chi-square test statistic?

What is the critical value of the chi-square test statistic? Use alpha=0.05

What is the managerial decision for this test?

Answer 1

1) expected value for cell 1,2 =row total*colum total/grand total =37*32/100=11.84

2) observed value for cell 2,1 =36

3) applying chi square test:

Expected	Ei=row total*column total/grand total	Purcahsed	Not purcahsed	Total
	Men	20.16	11.84	32
	Women	42.84	25.16	68
	total	63	37	100
chi square    χ2	=(Oi-Ei)2/Ei	Purcahsed	Not purcahsed	Total
	Men	2.3207	3.9515	6.272
	Women	1.0921	1.8595	2.952
	total	3.413	5.811	9.2238

calculated value of the chi-square test statistic =9.2238

4) critical value of the chi-square test statistic =3.841

5) as test statisitc is in critical region we can conclude that response to the question of whether they have purchased anything over the Internet is dependent of gender.

Answer 2

To answer these questions, let's calculate the expected value for cell 1,2, the observed value for cell 2,1, the chi-square test statistic, and the critical value of the chi-square test statistic.

Step 1: Calculate the expected value for cell 1,2.

The expected value for a cell in a 2x2 contingency table can be calculated using the formula: Expected Value = (Row Total * Column Total) / Grand Total

For cell 1,2: Expected Value = (Row 1 Total * Column 2 Total) / Grand Total Expected Value = (32 * 37) / 100 Expected Value = 11.84 (rounded to two decimal places)

Step 2: Calculate the observed value for cell 2,1.

The observed value for cell 2,1 is already given in the table as 36.

Step 3: Calculate the chi-square test statistic.

The chi-square test statistic formula for a 2x2 contingency table is: χ^2 = Σ [(Observed Value - Expected Value)^2 / Expected Value]

For our table:

χ^2 = [(27 - 11.84)^2 / 11.84] + [(5 - 25.16)^2 / 25.16] + [(36 - 20.16)^2 / 20.16] + [(32 - 47.84)^2 / 47.84]

χ^2 ≈ 10.25 (rounded to two decimal places)

Step 4: Calculate the critical value of the chi-square test statistic.

The critical value of the chi-square test statistic can be found using a chi-square distribution table. For alpha = 0.05 and degrees of freedom (df) = (Number of rows - 1) * (Number of columns - 1) = (2 - 1) * (2 - 1) = 1, the critical value is approximately 3.841.

Step 5: Make the managerial decision.

To make the managerial decision, we compare the calculated chi-square test statistic with the critical value. If the calculated value is greater than the critical value, we reject the null hypothesis (which states that the response to the question of whether they have purchased anything over the Internet is independent of gender). If the calculated value is less than or equal to the critical value, we fail to reject the null hypothesis.

In this case, the calculated chi-square test statistic (10.25) is greater than the critical value (3.841). Therefore, we reject the null hypothesis, and we can conclude that the response to the question of whether individuals have purchased anything over the Internet is not independent of gender. There is a significant relationship between gender and Internet buying behavior.