Question

One of the products produced by Branco Food Company is All-Bran Cereal, which competes with three other brands of similar all-bran cereals. The company's research office wants to investigate if the percentage of people who consume all-bran cereal is the same for each of these four brands. Let us denote the four brands of cereal by A,B,C, and D. A sample of 900 persons who consume all-bran cereal was taken, and they were asked which brand they most often consume. Of the respondents, 226said they usually consume Brand A, 232 consume Brand B, 233 consume Brand C, and 209 consume Brand D. Does the sample provide enough evidence to reject the null hypothesis that the percentage of people who consume all-bran cereal is the same for all four brands? Use α=0.05.

Answer 1

Solution

Solution is based on Chi-square Test for Goodness of Fit.

Let p_A, p_B, p_C and p_D be the proportion of people who consume all-bran cereal brands A, B, C and D respectively.

Let Oi and Ei be respectively the observed and expected frequencies of the ith class, i = 1 to 4, 4 being the number of brands given.

Claim: the percentage of people who consume all-bran cereal is the same for all four brands

Hypotheses:

Null: H₀: p_A = p_B = p_C = p_D [claim]

Vs

Alternative H_A: H₀ is false, i.e., at least one of p_A, p_B, p_C and p_D is different from others.

Test Statistic:

χ2 = ∑[i = 1,k]{(Oi - Ei)²/Ei},

Under H₀, Ei must be the same for A, B, C and D and hence each must be equal to total number of respondents/4 = 225. With the above terminology, χ²_cal = 1.6444

Details of calculations

Brand	A	B	C	D	Total
Oi	226	232	233	209	900
Ei	225	225	225	225	900
χ2	0.0044444	0.2177778	0.284444	1.13777778	1.644444
CHECK: ΣOi = ΣEi.		Done	k	4

χ2cal

1.6444444

DF

4

χ2crit

9.487729

p-value

0.80078285

Distribution, Significance Level, α, Critical Value, p-value

Under H0, χ² ~ χ²_{k – s}, Chi-square distribution with degrees of freedom = k – s,where k = number of classes and s =number of parameters estimated.

p-value = P(χ²_{k – s} > χ²_cal) [here k = 4 and s = 0]

Given significance level = α , critical value = χ²_crit = upper α% of χ²_{k - s, α}

Critical value and p-value obtained using Excel Function: Statistical CHIINV and CHIDIST are as shown in the above table.

Decision

Since, χ2_cal < χ2_crit, or equivalently, since p-value > α, H₀ is accepted

Conclusion

There is sufficient statistical evidence in favor of the claim and hence we conclude that

the percentage of people who consume all-bran cereal is the same for all four brands. ANSWER

DONE