Question

sthed Chapter 11. Supplementary Exercises Problem 051 One of the products produced by Branco Food Company is Allan Cerest, which competes with three other brands of similar all bran cereals. The company's research office wants to investigate 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 1100 persons who consume all bran cereal was taken and they were asked which brand they most often consume of the respondents, 251 said they usually consume Brand A. 245 consume Brand B. 383 consume Brand C. and 221 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 a 0.1 Find the value of the test statisticx? Round your answer to three decimal places thes

Answer 1

$Let~p_i=true~proportion~of~people~who~consume~all~bran~cereal\\~for~ith~brand;~i=A,B,C,D.\\ Null~hypothesis,~H_0:~p_1=p_2=p_3=p_4=1/4\\ vs.~Alternative~hypothesis,~H_a:~At~least~two~p_i~\neq 1/4.\\ Observed~frequencies:~O_A=251,~O_B=245,~O_C=383,~O_D=221\\ Expected~frequencies:~E_A=E_B=E_C=E_D=1100/4= 275\\ Value~of~test~statistic=\chi^2=\sum \frac{(O_i-E_i)^2}{E_i}\\ =(251-275)^2/275&plus;(245-275)^2/275&plus;(383-275)^2/275&plus;(221-275)^2/275\\ =58.385.\\ p-value=P(\chi^2>58.385|\chi^2\sim \chi^2_{3})=0.0000<0.1.\\ Hence~we~reject~H_0~at~level~0.1~and~conclude~that~there~is~enough~evidence\\ that~the~percentage~of~people~who~consume~all~bran~cereal~is~not~same~for~all~four\\ brands.$