Question

Researchers want to determine whether all bags of M&Ms® have the same proportion of colors regardless of the flavor of M&Ms®. To test this, they sampled randomly king-size bags of each flavor and recorded their findings in the table.

Flavor	M&M's® Color
	Red	Orange	Yellow	Green	Blue	Brown
Original	24	11	29	17	9	14
Peanut	15	20	30	25	15	19
Almond	22	17	21	12	28	7

Part A: What are the correct degrees of freedom for this table? (2 points)

Part B: Calculate the expected count for the number of green peanut M&Ms®. Show your work. (3 points)

Part C: Is there sufficient evidence that there is a difference in the proportion of colors for the different flavors of M&Ms®? Provide a statistical justification for your conclusion. (5 points)

Answer 1

The Observed and Expected value tables are as below. Each cell of the expected value = (Row Total * Column Total) / Total

Observed
Race	Red	Orange	Yellow	Green	Blue	Brown	Total
Original	24	11	29	17	9	14	104
Peanut	15	20	30	25	15	19	124
Almond	22	17	21	12	28	7	107
Total	61	48	80	54	52	40	335

Expected
Race	Red	Orange	Yellow	Green	Blue	Brown	Total
Original	18.937	14.901	24.836	16.764	16.143	12.418	92
Peanut	22.579	17.767	29.612	19.988	19.248	14.806	109
Almond	19.484	15.331	25.552	17.248	16.609	12.776	94
Total	61	48	80	54	52	40	335

(A) Degrees of freedom = (Row -1) * (Column - 1) = (3 - 1) * (6 - 1) = 2 * 5 = 10

(B) Expected value = 54 * 124 / 335 = 19.988

(C)

The Hypothesis:

H0: The sample has a distribution that agrees with the distribution of the populations.

Ha: The sample has a distribution that is different from the distribution of the populations.

The Test Statistic:

	Observed	Expected	O-E	(O-E)2	(O-E)2/E
1	24	18.937	5.063	25.631	1.353
2	15	22.579	-7.579	57.443	2.544
3	22	19.484	2.516	6.332	0.325
4	11	14.901	-3.901	15.222	1.021
5	20	17.767	2.233	4.986	0.281
6	17	15.331	1.669	2.784	0.182
7	29	24.836	4.164	17.340	0.698
8	30	29.612	0.388	0.151	0.005
9	21	25.552	-4.552	20.723	0.811
10	17	16.764	0.236	0.056	0.003
11	25	19.988	5.012	25.120	1.257
12	12	17.248	-5.248	27.539	1.597
13	9	16.143	-7.143	51.027	3.161
14	15	19.248	-4.248	18.043	0.937
15	28	16.609	11.391	129.756	7.812
16	14	12.418	1.582	2.503	0.202
17	19	14.806	4.194	17.590	1.188
18	7	12.78	-5.776	33.364	2.611
Total					25.989

test = 25.989

The Critical Value:   The critical value at = 0.05 (default) , df= n - 1 = 10

critical = 18.307

The p value: The p value: The p value at test = 25.989, df = 10; P value = 0.0038

The Decision Rule:

The Critical Value Method: If test is > critical, then Reject H0.

The p - value Method: If p value is < , Then Reject H0.

The Decision:  

The Critical Value Method: Since test (25.989) is > critical (18.307), We Reject H0.

The p - value Method: Since p value (0.0038) is < (0.05), We Reject H0.

The Conclusion: There is sufficient evidence at the 95% significance level to conclude that there is a difference in proportion of colors for the different flavors of M & M's.