Question

A market researcher believes that brand perception of one of the company's products may vary between different groups. After interviewing 397 persons, the following data was compiled. Use the 0.05 level of significance.

Can we conclude that brand perception is dependent on age?

Age 18-30 30-45 over 45 TOTAL
Favorable 50 65 65 180
Unfavorable 18 22 22 62
Neutral 19 24 24 67

Answer 1

Explanation:

Hello!

The objective is to test if the brand perception of the company varies depending on the customer's age.

309 customers were interviewed and their age and opinion were registered, so you have two variables of interest:

X₁: Age of the customer. Categorized: "18-30", "30-45" and "over 45"

X₂: Perception of the company brand. Categorized: "Favorable", "Unfavorable" and "Neutral"

You have two categorical variables and need to study the association between them, the best statistic test to do in this case is a Chi-Square of Independence test, based on this, the hypotheses are:

H₀: Pij= Pi. * P.j ∀ i=1, 2, 3 and j=1, 2, 3

H₁: The variables X₁ and X₂ are dependent.

α: 0.05

The statistic is:

`X^2= double sum \frac{(O{ij}-E_(ij))^2}{E_(ij)} ~~X^2_((r-1)(c-1))`

Oij= observed frequency for the i- j- category

Eij= expected frequency for the i-j- category

r= total number of categories in the rows

c= total number of categories in the colums

Before calculating the value of
`X^2_(H_0)` you have to calculate the expected frequencies for each category:

`E_(ij)= \frac{Oi{_.}*O_(.j)}{n}`

Where Oi.= total of the i-row (marginal) and O.j= total of the j-column (marginal)

For example for the categories "favorable" and "18-30" the expected value will be:

`E_(11)= (O_(1.)*O_(.1))/(n) = (180*87)/(309)= 50.68`

(see attachment for full tables) Note, the sample size of the given data is 309, for the calculatios I've used that value.

`X^2_(H_0)= (((50-50.68)^2)/(50.68) )+(((65-64.66)^2)/(64.66) )+(((65-64.66)^2)/(64.66) )+(((18-17.46)^2)/(17.46) )+(((22-22.27)^2)/(22.27) )+(((22-22.27)^2)/(22.27) )+(((19-18.86)^2)/(18.86) )+(((24-24.07)^2)/(24.07) )+(((24-24.07)^2)/(24.07) )= 0.037`

This type of test is always one-tailed to the right and so is its p-value:

P(X²₄≥0.037)= 1 - P(X²₄<0.037)= 1 - 0.0002= 0.9998

The p-value= 0.9998 is greater than the significance level α: 0.05, then the decision is to not reject the null hypothesis:

Using a significance level of 5%, there is no significant evidence to reject the null hypothesis. You can conclude that the Age of the customers and their brand perception are independent.

I hope it helps!