Question

Researchers want to determine whether all frozen pizzas have the same proportion of toppings regardless of the brand of pizza. To test this, they sampled randomly pizzas of each brand and recorded their findings in the table.

Brand	Topping
	Pepperoni	Sausage	Onion	Mushrooms
Digiorno	14	21	17	17
Tombstone	11	8	10	6
Red Baron®	17	13	14	11
Tony's	9	17	15	18

Part A: What are the correct degrees of freedom for this table? (2 points)

Part B: Calculate the expected count for the number of pepperoni for Red Baron® pizza. Show your work. (3 points)

Part C: Is there sufficient evidence that there is a difference in the proportion of toppings for the different brands of pizza? Provide a statistical justification for your conclusion. (5 points)

A large toiletry distributor claims that 35% of all individuals who purchase toilet paper from the stores that carry its product choose original toilet paper, 28% choose sensitive toilet paper, 20% choose ultra-strong toilet paper, and 17% choose ultra-soft toilet paper. To investigate this claim, researchers collected data from a random sample of customers in a large city. The results were 170 packages of original, 105 sensitive, 80 ultra-strong, and 45 ultra-soft toilet paper purchases. Are the data from the sample consistent with the distributor's claim? Conduct an appropriate statistical test at the 5% significance level to support your conclusion. Make sure to include parameters, check conditions, and show calculations before formulating a conclusion. (10 points)

Answer 1

Part A:

Degree of freedom: df =( number of rows -1)*(number of columns-1) = (4-1)*(4-1)=9

Part B:

Following table shows the row total and column total:

Brand	Topping
	Pepperoni	Sausage	Onion	Mushrooms	Total
Digiorno	14	21	17	17	69
Tombstone	11	8	10	6	35
Red Baron®	17	13	14	11	55
Tony's	9	17	15	18	59
Total	51	59	56	52	218

The  expected count for the number of pepperoni for Red Baron® pizza is

Part C:

Following table shows the expected frequencies:

	Pepperoni	Sausage	Onion	Mushrooms	Total
Digiorno	16.142	18.674	17.725	16.459	69
Tombstone	8.188	9.472	8.991	8.349	35
Red Baron®	12.867	14.885	14.128	13.119	55
Tony's	13.803	15.968	15.156	14.073	59
Total	51	59	56	52	218

Following table shows the calculations for chi square test statistics:

O	E	(O-E)^2/E
14	16.142	0.284237641
11	8.188	0.965723498
17	12.867	1.327558017
9	13.803	1.671289502
21	18.674	0.289722395
8	9.472	0.228756757
13	14.885	0.23871179
17	15.968	0.066697395
17	17.725	0.029654443
10	8.991	0.113233344
14	14.128	0.001159683
15	15.156	0.001605701
17	16.459	0.017782429
6	8.349	0.66089364
11	13.119	0.342263968
18	14.073	1.095809635
Total		7.335099838

Following is the test statistics:

The p-value is: 0.6023

Since p-value is greater than 0.05 so we fail to reject the null hypothesis. There is not  sufficient evidence that there is a difference in the proportion of toppings for the different brands of pizza.