Question

Use the distribution below to test that Plain M&Ms follow the stated distribution. Discuss your choice of ?. Would a different ? have changed your conclusion?

M&M states the following distribution for Plain M&Ms:
Red = 13%, Orange = 20%, Yellow = 13%, Green = 20%, Blue = 20%, Brown = 14%

Total number of plain M&M's: 665 Total number of red plain M&M's: 76 Total number of plain brown M&M's: 66 Total number of blue plain M&M's: 179 Total number of orange plain M&M's: 140 Total number of green plain M&M's: 119  Total number of yellow plain M&M's: 85

Total number of peanut M&M's: 356 Total number of red peanut M&M's: 22 Total number of peanut brown M&M's: 34 Total number of blue peanut M&M's: 55 Total number of orange peanut M&M's: 62 Total number of green peanut M&M's: 72  Total number of yellow peanut M&M's: 111

Answer 1

Chi-square goodness of fit test:

Number of categories, n =6

Null Hypothesis(H0):

Plain M&Ms follow the stated distribution.

Alternative Hypothesis(H1):

Plain M&Ms do not follow the stated distribution.

Calculation of the test statistic:

Color of plain M&M	Observed frequencies: O	Expected frequencies: E	(O - E)2/E
Red	76	665*13% =86.45	1.2632
Orange	140	665*20% =133	0.3684
Yellow	85	665*13% =86.45	0.0243
Green	119	665*20% =133	1.4737
Blue	179	665*20% =133	15.9098
Brown	66	665*14% =93.1	7.8884
Total	665	665	26.9278

Test statistic, 26.9278

Comparing with the critical value:

Let the significance level, =0.05

It means, the probability of rejecting the true null hypothesis =P(type 1 error) =0.05

Degrees of freedom, df =n - 1 =6 - 1 =5

The critical value is: =11.0705

Decision criteria:

Reject H0 if ​​​​​​. Otherwise, do not reject it.

Conclusion:

26.9278 > 11.0705

Since ​​​​​​, reject the null hypothesis(H0) at 5% significance level.

Thus, there is a sufficient statistical evidence at 5% significance level to claim that the Plain M&Ms do not follow the stated distribution.

Let us take different significance level of 0.01.

At =0.01 and df =5, =15.0863

The conclusion remains the same at 0.01 significance level because 26.9278 > 15.0863