Question

company A was 45%, whereas company B had 40% of the market. Other competitors accounted for the remaining 15%.To determine whether these market shares changed after the advertising campaigns, a marketing analyst solicited the preferences of a random sample of 200 customers of fabric softener. Of the 200 customers, 102 indicated a preference for company A's product, 82 preferred company B's fabric softener, and the remaining 16 preferred the products of one of the competitors. Can the analyst infer at the 5% significance level that customer preferences have changed from their levels before the advertising campaigns were launched?

Answer 1

Answer :

it is an example of chi square test :

Observed frequency :

Company A 102

Company B 82

Other 16

Total = N = 200

We have to do hypothesis testing using chi square test :

here proportion of Company A = 45 % i = 0.45

for company B = 40 % = 0.40 and other company = 15 % = 0.15

## To test :  

Ho : P₁ = 0.45 , P₂ = 0.40 and P₃ = 0.15

VS

H1 : At least one P_i   is not equal to its specified value .

## Test statistics :

χ2 = Σ ( F_i - E_i )² / E_i

Where   F_i = Observed Frequency and

E_i    = Expected frequency .

Where E_i   = N * P_i

E₁ = Expected frequency for company A = 200 * 0.45 = 90

E₂ = Expected frequency for company B = 200 * 0.40 = 80

E₃ = Expected frequency for other company    = 200 * 0.15 = 30

Company	Fi	Ei	( Fi - Ei)	( Fi - Ei) 2	( Fi - Ei) 2 / Ei
A	102	90	12	144	1.60
B	82	80	2	4	0.05
other	16	30	- 14	196	6.53
Total	200	200	0	344	8.18

χ2 = 8.18

# DF = degree of freedom = k -1 here k = 3

DF = 2

α = 0.05 level of significance .

# Chi square critical value = χ2 ( α , 0.05) = χ2 (0.05 ,2 )

= 5.99 ( from table )

## Decision :

We reject Ho if χ2 statistics value >     χ2 (0.05 ,2 ) critical value .

here χ2 statistics value >     χ2 (0.05 ,2 ) we reject Ho at given level of significance .

## Conclusion :

There is sufficient evidence to conclude that or infer that the proportion have changed since the advertising compaigns were implemented