Question

You are the market researcher for Chili’s. Your boss wants to find out if the restaurant enjoys the same market penetration in the households with children as in households without children. He asked you, the market researcher, to look into this problem. You decide to take a random sample of 350 households with children and find that 45% of them eat at Chili’s. Then, you take a random sample of 250 households without children and find out that 54% of them eat at Chili’s. Given this, does the restaurant have the same or different market penetrations among the two market segments? To answer this question, please show your detailed work by following the steps below.

a.) What are the hypotheses (null and alternative) you test? Describe in words what you test. (2 points)

b.) Is the market penetration the same among the two market segments? The confidence level is 95%. (3 points)

c.) What is the conclusion, that is, what do you suggest your boss to do with the information gathered from this analysis? Be clear in your recommendation and explain why you reached it.(3 points)

Answer 1

Answer a)

The following null and alternative hypotheses need to be tested:

Ho: p1​ = p2​

Ha: p1​ ≠ p2​

p1= True proportion of households with children that eat at Chili's

p1= True proportion of households without children that eat at Chili's

Answer b)

p1 = 0.45, n1 = 350

p2 = 0.54, n2 = 250

Pooled Proportion = (0.45*350+0.54*250)/(350+250) = 0.4875

Test Statistic

z = (0.45-0.54)/sqrt(0.4875*(1-0.4875)*(1/350+1/250))

Test statistic z = -2.17

Critical Value

Based on the information provided, the significance level is \alpha = 0.05α=0.05, and the critical value for a two-tailed test is zc​ = 1.96

Decision

Since it is observed that |z| = 2.14 > zc​ = 1.96, it is then concluded that the null hypothesis is rejected.

Conclusion

At 5% significance level, there is enough evidence to reject the claim that the market penetration the same among the two market segments.

Answer c)

Based on the infomration gathered and analysis done, I would inform boss that the restaurant does not enjoys the same market penetration in the households with children as in households without children. The proprotion of households without children that eats at Chili's is more than the  proprotion of households without children that eats at Chili's.