Question

In an effort to make childrenâs toys safer and more tamperproof, toy packaging has become cumbersome for parents to remove in many cases. Accordingly, the director of marketing at Toys4Tots, a large toy manufacturer, wants to evaluate the effectiveness of a new packaging design that engineers claim will reduce customer complaints by more than 10 percentage points. Customer satisfaction surveys were sent to 220 parents who registered toys packaged under the old design and 220 parents who registered toys packaged under the new design. Of these, 81 parents expressed dissatisfaction with packaging of the old design, and 41 parents expressed dissatisfaction with packaging of the new design.

Let p1 represent the population proportion of parents that expressed dissatisfaction with the packaging of the old design and p2 represent the population proportion of parents that expressed dissatisfaction with the packaging of the new design.

Required:
a. Specify the null and alternative hypotheses to test for whether the proportion of customer complaints has decreased by more than 5% under the new packaging design.
b. What is the value of the test statistic and the associated p-value?
c. At the 1% significance level, do the results support the engineers' claim?
d. At the 10% significance level, do the results support the engineers' claim?

Answer 1

Kindly check explanation

Explanation:

Given the following :

Old design samples(n1) = 220

New design samples (n2) = 220

P1 = 81/220 = 0.368 ; P2 = 41/220 = 0.186

Null hypothesis : p1 - p2 ≥ 0.05

Alternative hypothesis : p1 - p2 < 0.05

Using the test statistic formula :

t = (P1 - P2) / √[P(1-P)(1/n1 + 1/n2)]

P = (81 + 41)/(220 + 220) = 122/440 = 0.277

1 - P = 0.723

t = (0.368 - 0.186) / (√(0.277(0.723)(1/220 + 1/220)))

t = 4.265

P value from z test at α = 0.05

P value = 0.00002

Test for significance at α = 0.01 ;

Test for significance at α = 0.1

Result is significant at both α levels, hence reject the Null.