Question

Caption:Princess Foods Corporation has observed the changing awareness of the population on health and nutrition. Therefore, they want to investigate the acceptance of a low-calorie product and a low-sodium product by market segment.(gender) Are people more concerned about low-calorie soups or low-sodium soups and how does that break down by market segment (age)?

Mieke:Here’s what we did: Two hundred customers were selected at random for two different interviews. We were hoping that the information that we gleaned from these interviews would indicate the relative interest in low-calorie and/or low-sodium soup and how that interest was broken down by market segment. That’s going to give us insight into what the market wants and insight into who this customer is.

“Which of the following three products are you most interested in?” Then the results were tallied, indicating by age category preference for each of the three options. A study for each collected the following data. Test for independence at a significance of 5%.

Categories	50 years or younger	Over 50
Low Sodium	31	40
Regular Broth	33	38
Creamed Soups	36	22

Answer 1

Solution:

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

Null hypothesis: H0: Two variables are independent.

Alternative hypothesis: Ha: Two variables are dependent.

We assume level of significance = α = 0.05

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 = 3

Number of columns = c = 2

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

α = 0.05

Critical value = 5.99146455

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Age
Categories	50 or less	Over 50	Total
Low sodium	31	40	71
Regular Broth	33	38	71
Creamed Soups	36	22	58
Total	100	100	200

Expected Frequencies
	Age
Categories	50 or less	Over 50	Total
Low sodium	35.5	35.5	71
Regular Broth	35.5	35.5	71
Creamed Soups	29	29	58
Total	100	100	200

Calculations
(O - E)
-4.5	4.5
-2.5	2.5
7	-7

(O - E)^2/E
0.570423	0.570423
0.176056	0.176056
1.689655	1.689655

Test Statistic = Chi square = ∑[(O – E)^2/E] = 4.87226809

χ2 statistic = 4.87226809

P-value = 0.08749846

(By using Chi square table or excel)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is sufficient evidence to conclude that the two variables age categories and product categories are independent.