Question

Suppose a candy company representative claims that colored candies are mixed such that each large production batch has precisely the following proportions: 10% brown, 10% yellow, 30% red, 10% orange, 20% green, and 20%20% blue. The colors present in a sample of 410 candies was recorded. Is the representative's claim about the expected proportions of each color refuted by the data?

Color   Number of Candies
brown   80
yellow   101
red   70
orange   60
green   39
blue   60

State the null and alternative hypothesis.

What does the null hypothesis indicate about the proportions of candies of each color?

State the null and alternative hypothesis in terms of the expected proportions for each category.

Find the expected value for the number of chocolate candies colored brown. Round your answer to two decimal places.

Find the expected value for the number of chocolate candies colored yellow. Round your answer to two decimal places.

Find the value of the test statistic. Round your answer to three decimal places

Find the degrees of freedom associated with the test statistic for this problem.

Find the critical value of the test at the 0.010.01 level of significance. Round your answer to three decimal places.

Make the decision to reject or fail to reject the null hypothesis at the 0.010.01 level of significance.

State the conclusion of the hypothesis test at the 0.010.01 level of significance.

Answer 1

Category	Observed Frequency (O)	Proportion, p	Expected Frequency (E)	(O-E)²/E
Brown	80	0.1	410 * 0.1 = 41	(80 - 41)²/41 = 37.0976
Yellow	101	0.1	410 * 0.1 = 41	(101 - 41)²/41 = 87.8049
Red	70	0.3	410 * 0.3 = 123	(70 - 123)²/123 = 22.8374
Orange	60	0.1	410 * 0.1 = 41	(60 - 41)²/41 = 8.8049
Green	39	0.2	410 * 0.2 = 82	(39 - 82)²/82 = 22.5488
blue	60	0.2	410 * 0.2 = 82	(60 - 82)²/82 = 5.9024
Total	410	1.00	410	184.996

Null and Alternative hypothesis:

Ho: Proportions are same as expected.

H1: Proportions are different.

Expected value for the number of chocolate candies colored brown = 410 * 0.1 = 41

Expected value for the number of chocolate candies colored yellow = 410 * 0.1 = 41

Test statistic:

χ² = ∑ ((O-E)²/E) = 184.996

df = n-1 = 5

Critical value:

χ²α = CHISQ.INV.RT(0.01, 5) = 15.086

Decision:

Reject the null hypothesis

Conclusion:

There is enough evidence to conclude that proportions are different at 0.01 significance level.