Question

1. Old Yellow produces cereal and other food items with a focus on providing healthy and delicious food at an affordable price. In the face of growing competition in the health food cereal market, Old Yellow is considering redesigning the package of its most popular cereal. Two potential new designs have been developed: Design A and Design B.

   Before a final decision is made on which (if either) of the new designs to use, a focus group is conducted to measure consumer opinions and preferences for the 3 designs. A sample of 80 consumers is asked to review the 3 designs, provide specific comments on what they liked or didn’t like about each design, and state which package design they liked best. The results of design preference (1=current design; 2= ‘new design A’; 3= ‘new design B’) were recorded. A partial data set is provided below

   You have been asked to analyze the results to determine if there is any difference in preference for ‘new design A’ between younger consumers (under 55) and older consumers (55 and older).

   Younger Consumers	Older Consumers
   3	2
   2	3
   1	2
   2	1
   2	2
   …	…
   …	…
   …	…

   F-Test Two-Sample for Variances
   	Younger Consumers	Older Consumers
   Mean	2	1.675
   Variance	0.5641	0.2763
   Observations	40	40
   df	39	39
   F	2.0418
   P(F<=f) one-tail	0.0142
   F Critical one-tail	1.5137

   t-Test: Paired Two Sample for Means
   	Younger Consumers	Older Consumers
   Mean	2	1.675
   Variance	0.5641	0.2763
   Observations	40	40
   Pearson Correlation	0.4547
   Hypothesized Mean Difference	0
   df	39
   t Stat	2.9625
   P(T<=t) one-tail	0.0026
   t Critical one-tail	1.3036
   P(T<=t) two-tail	0.0052
   t Critical two-tail	1.6849

   t-Test: Two-Sample Assuming Equal Variances
   	Younger Consumers	Older Consumers
   Mean	2	1.675
   Variance	0.5641	0.2763
   Observations	40	40
   Pooled Variance	0.4202
   Hypothesized Mean Difference	0
   df	78
   t Stat	2.2422
   P(T<=t) one-tail	0.0139
   t Critical one-tail	1.2925
   P(T<=t) two-tail	0.0278
   t Critical two-tail	1.6646

   t-Test: Two-Sample Assuming Unequal Variances
   	Younger Consumers	Older Consumers
   Mean	2	1.675
   Variance	0.5641	0.2763
   Observations	40	40
   Hypothesized Mean Difference	0
   df	70
   t Stat	2.2422
   P(T<=t) one-tail	0.0141
   t Critical one-tail	1.2938
   P(T<=t) two-tail	0.0281
   t Critical two-tail	1.6669

   z-Test: Two Proportions
   	Younger Consumers	Older Consumers
   Sample Proportions	0.45	0.625
   Observations	40	40
   Hypothesized Difference	0
   z Stat	-1.5697
   P(Z<=z) one tail	0.0582
   z Critical one-tail	1.2816
   P(Z<=z) two-tail	0.1164
   z Critical two-tail	1.6449

   Based on the correct output above, which of the following decisions should be made (α = 0.10)?

   		None of these are correct
   		Fail to reject H0
   		You cannot use α = 0.10
   		Reject H0

Answer 1

The following are determined from the given statements

(i) We have 2 different populations, Younger consumers and Older: Therefore this will be a 2 independent samples test.

(ii) We need to find if there is a difference in preference (the term variability is not mentioned): So this is a 2 tailed test for the mean.

(iii) Since population variance is unknown, we will do a t test

(iv) Since s1/s2 = Sqrt(0.564 / 0.2763) = 1.43 which is < 2, we use the pooled Variance.

Therefore we use a 2 tailed 2 independent samples t test assuming equal variances

The Decision rule is that if p value is < , then reject H0.

The 2 tailed p value = 0.0278 which is < (0.10)

Therefore Option 4: Reject H0