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, and 16% 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

Color	Frequency	Claimed Proportion
Brown	60	0.13
Yellow	67	0.14
Red	52	0.13
Blue	60	0.24
Orange	78	0.20
Green	66	0.16

Determine the null and alternative hypotheses. Choose the correct answer below.

A.H0​:The distribution of colors is not the same as stated by the manufacturer.

H1​:The distribution of colors is the same as stated by the manufacturer.

B.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.

C.None of these.

Compute the expected counts for each color.

Color	Frequency	Expected Count
Brown	60 ?
Yellow	67                     ?
Red	52 ?
Blue	60 ?
Orange	78 ?
Green	66 ?

chi Subscript 0 Superscript 2=?

What is the​ P-value of the​ test?

​P-value=?

Based on the​ results, do the colors follow the same distribution as stated in the​ problem?

A.Do not reject Upper H 0. There is not sufficient evidence that the distribution of colors is not the same as stated by the manufacturer.

B.Do not reject Upper H 0. There is sufficient evidence that the distribution of colors is not the same as stated by the manufacturer.

C.Reject Upper H 0. There is not sufficient evidence that the distribution of colors is not the same as stated by the manufacturer.

D.Reject Upper H 0. There is sufficient evidence that the distribution of colors is not the same as stated by the manufacturer.

Answer 1

Solution:

Given:

Color	Frequency	Claimed Proportion
Brown	60	0.13
Yellow	67	0.14
Red	52	0.13
Blue	60	0.24
Orange	78	0.20
Green	66	0.16

Part a) Determine the null and alternative hypotheses.

B. 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.

Part b) Compute the expected counts for each color.

Multiply each claimed proportion by N = 383.

Thus we get:

Color	Frequency	Expected Count
Brown	60	49.79
Yellow	67	53.62
Red	52	49.79
Blue	60	91.92
Orange	78	76.60
Green	66	61.28

Part c) 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	60	49.79	72.304
Yellow	67	53.62	83.719
Red	52	49.79	54.308
Blue	60	91.92	39.164
Orange	78	76.60	79.426
Green	66	61.28	71.084
	383

Thus

Part d) What is the​ P-value of the​ test?

Use following Excel command:

=CHISQ.DIST.RT( X² , df )

where

df = k - 1 = 6 - 1 = 5

=CHISQ.DIST.RT(17.004,5)

=0.004492

=0.0045

Thus P-value = 0.0045

Part e) Based on the​ results, do the colors follow the same distribution as stated in the​ problem?

Since P-value = 0.0045 < α = 0.05 level of significance, we reject the null hypothesis H0.

Thus correct answer is:

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