Question

A manufacturer of colored candies states that 13​%of the candies in a bag should be​ brown,14​%​yellow,13​%​red,24​%​blue,20​% ​orange, and16% green. A student randomly selected a bag of colored candies. He counted the number of candies of each color and obtained the results shown in the table. Test whether the bag of colored candies follows the distribution stated above at the alpha equalsα=0.05 level of significance.

Colored Candies in a bag

Color	Brown	Yellow	Red	Blue	Orange	Green
Frequency	61	67	57	59	96	66
Claimed Proportion	0.13	0.14	0.13	0.24	0.20	0.16

Please answer the expected count

Compute the expected counts for each color.

Color	Frequency	Expected Count
Brown	61
Yellow	67
Red	57
Blue	59
Orange	96
Green	66

Please answer the

test statistic

p-value

Reject or do reject

Thanks

Answer 1

Solution:

Given:

Color	Oi: Observed Frequency	Claimed Proportion
Brown	61	0.13
Yellow	67	0.14
Red	57	0.13
Blue	59	0.24
Orange	96	0.20
Green	66	0.16
	N= 406

State the hypothesis:

H₀​:The distribution of colors is the same as stated by the manufacturer.

H₁​:The distribution of colors is not the same as stated by the manufacturer.

Compute the expected counts for each color.

Multiply each claimed proportion by N = 406.

Thus we get:

Color	Oi: Observed Frequency	Ei: Expected Count
Brown	61	52.78
Yellow	67	56.84
Red	57	52.78
Blue	59	97.44
Orange	96	81.20
Green	66	64.96
	406

Find

Chi square test statistic for goodness of fit

Where

O_i = Observed Counts

E_i =Expected Counts

Thus we need to make following table:

Color	Oi: Observed Frequency	Ei: Expected Count	Oi2/Ei
Brown	61	52.78	70.500
Yellow	67	56.84	78.976
Red	57	52.78	61.557
Blue	59	97.44	35.725
Orange	96	81.20	113.498
Green	66	64.96	67.057
	N = 406

Thus

Find p-value:

Use following Excel command:

=CHISQ.DIST.RT( X² , df )

where

df = k - 1 = 6 - 1 = 5

=CHISQ.DIST.RT(21.312, 5)

=0.0007

Thus p-value = 0.0007

Decision: Reject or do not reject

Since P-value = 0.0007< α = 0.05 level of significance, we reject the null hypothesis H0.

Thus correct answer is:

Reject H₀. There is sufficient evidence that the distribution of colors is not the same as stated by the manufacturer.