Our population of interest is all 0.9 oz bags of Welch’s Mixed Fruit snacks.  We needed a large sample of fruit snacks in order to assure that our proportion tests would be valid and to try to procure the most accurate results possible.  To obtain our sample, we went to Costco in Avon, Ohio and bought an 80 count box of 0.9 oz bags of Welch’s Mixed Fruit Snacks.  We used all 80 bags in our study and counted the amount of each flavor in each bag, resulting in a total of 967 fruit snacks. 

The first test we used was a χ² Goodness of Fit Test because we wanted to see if the proportions of fruit snack flavors for the five different flavors were the same.

π1= the true proportion of grape fruit snacks; π2 = the true proportion of strawberry fruit snacks; π3 = the true proportion of peach fruit snacks; π4 = the true proportion of orange fruit snacks; π5 = the true proportion of raspberry fruit snacks.

H_o π1= 0.2, π2 = 0.2, π3 = 0.2, π4 = 0.2, π5 = 0.2

H_a : H_o  is not true

α = 0.05

Assumptions:  The sample was randomly chosen and all expected cell counts are greater than 5, so the sample size is large enough to carry out a Chi-Squared Goodness of Fit Test.

	Grape	Strawberry	Peach	Orange	Raspberry
Observed	109	286	287	132	153
Expected	193.4	193.4	193.4	193.4	193.4

 

χ² = 154.4012

P-value = 0 with 4 degrees of freedom

We are using a two proportion  Z-test to compare the proportion of grape fruit snacks to strawberry fruit snacks and then peach and strawberry fruit snacks.

π1= the true proportion of grape fruit snacks in a 0.9 oz bag

π2= the true proportion of strawberry fruit snacks in a 0.9 oz bag

H_{o :} π1- π 2=0

H_a : π1- π 2-  ≠  0

α=0.05

  Assumptions: Both samples are random and large enough since:

n1(π1) ≥ 10         n1(1- π1) ≥ 10          n2(π2) ≥ 10                 n2(1- π2) ≥ 10

967(.11) ≥ 10      967(.89) ≥ 10           967(.30) ≥ 10            967(.70) ≥ 10

  

Z=-9.9835188

P=0

Peach and Strawberry

π1= the true proportion of strawberry fruit snacks in a 0.9 oz bag

π2 = the true proportion of peach fruit snacks in a 0.9 oz bag

H_{o :} π1- π 2=0

H_a : π1- π 2-  ≠  0

α=0.05

Assumptions: Both samples are random and large enough since:

n1(π1) ≥ 10         n1(1- π1) ≥ 10          n2(π2) ≥ 10           n2(1- π2) ≥ 10

967(.30) ≥ 10      967(.70) ≥ 10           967(.30) ≥ 10        967(.70) ≥ 10

  

Z=-0.04979914

P=0.9602823