Question

Mars Inc. claims that they produce M&Ms with the following distributions: Brown 20% Red 25% Yellow 25% Orange 5% Green 15% Blue 10% A bag of M&Ms was randomly selected from the grocery store shelf, and the color counts were: Brown 21 Red 21 Yellow 19 Orange 12 Green 17 Blue 13 Using the χ2 goodness of fit test (α = 0.05) to determine if the proportion of M&Ms is what is claimed. Select the [test statistic, p-value, Decision to Reject (RH0) or Failure to Reject (FRH0)].

a) [χ2 = 12.628, p-value = 0.014, RH0]

b) [χ2 = 12.628, p-value = 0.027, RH0]

c) [χ2 = 6.314, p-value = 0.973, RH0]

d) [χ2 = 6.314, p-value = 0.973, FRH0]

e) [χ2 = 12.628, p-value = 0.027, FRH0]

f) None of the above

Answer 1

Solution:

Given:

Claim: Mars Inc. produce M&Ms with the following distributions:

Brown 20% Red 25% Yellow 25% Orange 5% Green 15% Blue 10%.

We have to use  the χ2 goodness of fit test (α = 0.05) to determine if the proportion of M&Ms is what is claimed.

Step 1) State H0 and H1:

H0: the proportion of M&Ms is according to claimed proportions.

Vs

H1: the proportion of M&Ms is different from claimed proportions.

Step 2) test statistic:

Chi square test statistic for goodness of fit

Where

O_i = Observed Counts

E_i =Expected Counts = N * % value = 103 * % value

Thus we need to make following table

Color	Oi: Observed Frequencies	Expected Percentage	Ei	Oi2/Ei
Brown	21	20%	20.6	21.408
Red	21	25%	25.75	17.126
Yellow	19	25%	25.75	14.019
Orange	12	5%	5.15	27.961
Green	17	15%	15.45	18.706
Blue	13	10%	10.3	16.408
	N = 103	100%

Thus

Step 3) Find p-value:

Use excel command:

=CHI.DIST.RT( _X² value , df)

where  _X² value = and df = k - 1 = 6 - 1 = 5

( k = number of categories = number of colors = 6)

Thus

=CHISQ.DIST.RT(12.628,5)

=0.02712

Thus p-value = 0.027

Step 4) Decision :

Reject H0, if P-value < 0.05 level of significance, otherwise we fail to reject H0

Since p-value =0.027 < 0.05 level of significance, we reject H0.

Thus correct answer is:

b) [χ2 = 12.628, p-value = 0.027, RH0]