Question

multiple choice 4 points state the null and alternative hypotheses. it has been rumored that the color distribution of m&ms is 30% brown, 20% yellow, 20% red, 10% orange, 10% blue, and 10% green. you suspect this rumor is incorrect and decide to count how many of each color are contained in a randomly chosen bag in order to perform a chi - square goodness - of - fit test at a 5% significance level. color number observed brown 27 yellow 16 red 21 orange 12 blue 9 green 15 total 100 state the null and alternative hypotheses. $h_0:p_{brown}=0.3,p_{yellow}=0.2,p_{red}=0.2,p_{orange}=0.1,p_{blue}=0.1,p_{green}=0.1$ $h_1:$ at least one of the stated proportions is not correct $h_0:p_{brown}=0.3,p_{yellow}=0.2,p_{red}=0.2,p_{orange}=0.1,p_{blue}=0.1,p_{green}=0.1$ $h_1:$ all of the stated proportions are not correct

Explanation:

Step1: Define null hypothesis

The null hypothesis \(H_0\) assumes the stated proportions are correct.
\(H_0: P_{brown}=0.3, P_{yellow}=0.2, P_{red}=0.2, P_{orange}=0.1, P_{blue}=0.1, P_{green}=0.1\)

Step2: Define alternative hypothesis

The alternative hypothesis \(H_a\) assumes the stated proportions are not correct.
\(H_a:\text{At least one of the stated proportions is not correct}\)

Answer:

\(H_0: P_{brown}=0.3, P_{yellow}=0.2, P_{red}=0.2, P_{orange}=0.1, P_{blue}=0.1, P_{green}=0.1\); \(H_a:\text{At least one of the stated proportions is not correct}\)