Questions: A student wanted to know if MM's follow their stated distribution. Here is what the company claims: Color Percentage ------ Brown 15% Yellow 12% Orange 20% Red 13% Green 16% Blue 24% A student takes a random sample of 600 MMs and finds: Color Number ------ Brown 78 Yellow 84 Orange 144 Red 90 Green 120 Blue 90 What type of test would you use to test that MM follows their stated distribution? - Test for Independence - Goodness of Fit - Test for Homogeneity

A student wanted to know if MM's follow their stated distribution. Here is what the company claims:

Color Percentage
------
Brown 15%
Yellow 12%
Orange 20%
Red 13%
Green 16%
Blue 24%

A student takes a random sample of 600 MMs and finds:

Color Number
------
Brown 78
Yellow 84
Orange 144
Red 90
Green 120
Blue 90

What type of test would you use to test that MM follows their stated distribution?

- Test for Independence
- Goodness of Fit
- Test for Homogeneity

Transcript text: A student wanted to know if $M \& M s$ follow their stated distribution. Here is what the company claims: \begin{tabular}{|c|c|} \hline Color & Percentage \\ \hline Brown & $15 \%$ \\ \hline Yellow & $12 \%$ \\ \hline Orange & $20 \%$ \\ \hline Red & $13 \%$ \\ \hline Green & $16 \%$ \\ \hline Blue & $24 \%$ \\ \hline \end{tabular} A student takes a random sample of 600 M\&Ms and finds: \begin{tabular}{|l|l|} \hline Color & Number \\ \hline Brown & 78 \\ \hline Yellow & 84 \\ \hline Orange & 144 \\ \hline Red & 90 \\ \hline Green & 120 \\ \hline Blue & 90 \\ \hline \end{tabular} What type of test would you use to test that M\&M follows their stated distribution? Test for Independence Goodness of Fit Test for Homogeneity

Solution

Solution Steps

To determine if the observed distribution of M&Ms matches the company's claimed distribution, we would use a Chi-Square Goodness of Fit test. This test compares the observed frequencies of categories to the expected frequencies, which are calculated based on the claimed percentages and the total sample size.

Step 1: Identify the Type of Test

To determine if the observed distribution of M&Ms matches the company's stated distribution, we need to compare the observed frequencies with the expected frequencies. This scenario is a classic example of a Goodness of Fit test, which is used to see if a sample matches a distribution.

Step 2: Calculate Expected Frequencies

The expected frequency for each color is calculated by multiplying the total number of M&Ms by the percentage claimed by the company.

Total number of M&Ms = 600

\[ \begin{align_} \text{Expected frequency for Brown} &= 600 \times 0.15 = 90 \\ \text{Expected frequency for Yellow} &= 600 \times 0.12 = 72 \\ \text{Expected frequency for Orange} &= 600 \times 0.20 = 120 \\ \text{Expected frequency for Red} &= 600 \times 0.13 = 78 \\ \text{Expected frequency for Green} &= 600 \times 0.16 = 96 \\ \text{Expected frequency for Blue} &= 600 \times 0.24 = 144 \\ \end{align_} \]

Step 3: Set Up the Hypotheses

Null Hypothesis ($H_0$): The observed distribution of M&Ms matches the company's stated distribution.
Alternative Hypothesis ($H_a$): The observed distribution of M&Ms does not match the company's stated distribution.