Question

A national youth organization sells six different kinds of cookies during its annual cookie campaign. A local leader is curious about whether national sales of the six kinds of cookies are uniformly distributed. He randomly selects the amounts of each kind of cookies sold by five youths and combines them into the observed data that follow. Use α = .05 to determine whether the data indicate that sales for these six kinds of cookies are uniformly distributed.

Kind of Cookie	Observed Frequency
Chocolate chip	185
Peanut butter	168
Cheese cracker	155
Lemon flavored	161
Chocolate mint	215
Vanilla filled	165

Appendix A Statistical Tables

(Round the intermediate values to 2 decimal places. Round your answer to 2 decimal places.)

Observed χ2 =  
Cookie Sales is                                                           not uniformly distributed or uniformly distributed by kind of cookie.

Answer 1

Solution:

The null and alternative hypotheses are as follows:

H_{​​​​​​0} : The cookie sales is uniformly distributed by kind of cookie.

H_{​​​​​​1} : The cookie sales is not uniformly distributed by kind of cookie.

To test the hypothesis we shall use chi-square test of goodness of fit. The test statistic is given as follows:

Where, O_{​​​​​​i}​​​'s are observed frequencies and e_{​​​​​​i}​​​'s are expected frequencies.

The expected frequencies are obtained as follows:

There are total 6 types of cookies. Hence, probability of sale of any type of cookie is 1/6.

Total observed frequency (N) = 185+168+155+161+215+165

Total observed frequency (N) = 1049

Since, we are testing for uniform distribution, therefore the expected frequency for each type of the cookies will be 1049/6 = 174.8333

O​​​​​​i	e​​​​​​i	(O​​​​​​i - e​​​​​​i)2/e​​​​​​i
185	174.8333	0.5912
168	174.8333	0.2671
155	174.8333	2.2499
161	174.8333	1.0945
215	174.8333	9.2280
165	174.8333	0.5531
	Total	13.9838

From the above table we get,

The value of the test statistic is 13.9838.

Degrees of freedom = (n - 1) = (6 - 1) = 5

The p-value for the test statistic is given as follows:

p-value = P(χ^{​​​​​​2} > value of the test statistic)

p-value = P(χ^{​​​​​​2} > 13.9838)

p-value = 0.0157

We make decision rule as follows:

If p-value is greater than the significance level, then we fail to reject the null hypothesis (H_{​​​​​​0}) at given significance level.

If p-value is less than the significance level, then we reject the null hypothesis (H_{​​​​​​0}) at given significance level.

We have, p-value = 0.0157 and significance level = 0.05

(0.0157 > 0.05)

Since, p-value is less than significance level of 0.05, therefore we shall reject the null hypothesis at 0.05 significance level.

Conclusion : At significance level of 0.05, there is enough evidence to conclude that Cookie Sales is not uniformly distributed by kind of cookie.