Question

In an effort to make children’s 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 which engineers claim will reduce customer complaints by more than 5 percentage points. Customer satisfaction surveys were sent to 230 parents who registered toys packaged under the old design and 230 parents who registered toys packaged under the new design. Of these, 84 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.

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.

	H0: p1 – p2 ≤ 0.05; HA: p1 – p2 > 0.05
	H0: p1 – p2 = 0.05; HA: p1 – p2 ≠ 0.05
	H0: p1 – p2 ≥ 0.05; HA: p1 – p2 < 0.05

b.	What is the value of the test statistic and the associated p-value? (Round intermediate calculations to at least 4 decimal places. Round "Test statistic" value to 2 decimal places and "p-value" to 4 decimal places.)

Test statistic
p-value

c.	At the 5% significance level, do the results support the engineers’ claim?

(Click to select)(Yes/No), the results (Click to select)(support/do not support) the engineer’s claim that the new package design reduces customer complaints by more than 5%.

d.	At the 10% significance level, do the results support the engineers’ claim?

(Click to select)(Yes/ No), the results (Click to select)(support/ do not support) the engineer’s claim that the new package design reduces customer complaints by more than 5%.

Answer 1

Ans:

a)

H₀: p₁ – p₂ ≤ 0.05;

H_A: p₁ – p₂ > 0.05

b)

sample proportion 1=84/230=0.3652

sample proportion 2=41/230=0.1783

pooled proportion=(84+41)/(230+230)=0.2717

Test statistic:

z=((0.3652-0.1783)-0.05)/sqrt(0.2717*(1-0.2717)*((1/230)+(1/230)))

z=3.30

p-value=P(z>3.30)=0.0005

c)As,p-value<0.05,we reject the null hypothesis.

Yes, the results support the engineer’s claim that the new package design reduces customer complaints by more than 5%.

d)As,p-value<0.1,we reject the null hypothesis.

Yes, the results support the engineer’s claim that the new package design reduces customer complaints by more than 5%.