Question

A company produces a bath soap. Because of market competition the company decided to introduce a more attractive packaging. The first design featured several bright colors to distinguish it from other brands. The second design was light green in color with just the company logo on it. As a test to determine which design was better, two comparable supermarkets were chosen. In one supermarket, the company’s soap was packaged in the first design and in the other supermarket the company’s soap was packaged in the second design. Each supermarket sold 5 brands of soap including the one belonging to the company of interest. The sales data for a 1 week period is recorded below:

	Brightly colored	Light green	Difference
Company’s soap sold	200	152	48
Other soap sold	650	698	-48
Total soap sold	850	850	850

a) What is the 80% confidence interval of the difference? Interpret your result(s).

b) Because the first design is more expensive, the management has decided to use this design only if it is better than the light green design. Is there sufficient evidence at the a=.05 level of significance to use the first design?

Set up the hypotheses. Make sure you clearly indicate the parameters as it is for “brightly colored design” or for “Light green design”.

c) Calculate the test statistic. Specify which test statistic you are finding (z or t).

d) Find the p-value using the appropriate table. Be sure to include any appropriate degrees of freedom. Draw the curve showing p-value, critical value(s), and rejection area.

e) Interpret the result. Make sure to include the weight of the evidence, the conclusion, and the significance level.

f) Suppose in a related study you wanted the 90% confidence interval for the proportion of the company’s brightly colored soap chosen to have a width of no more than 0.06. How large of a sample size would you need?

Answer 1

In the t-test, we compare the results between two mean groups and differences between the two groups.

so by using R the confidence interval for the difference is,

> t.test(Difference,conf.level = 0.80)

> 80 percent confidence interval:
-253.4746 820.1413

so confidence interval at 20%(100-80) level of significance is,

(-253.4746 , 820.1413)

so at 80% we can say that the differance between price of brightly colored soap and light green soap is greater than -243 and less than 820.

Testing of Hypothesis:

H0: The average cost of both designs are the same (1 = 2)

against,

H1: The average cost of the first design is higher than the second.(1 > 2)

That is this is a left tailed t-test.

so by using the R code at 5% level of significance:

> t.test(Brightly_colored,Light_Green,alternative = "greater",conf.level = 0.95)

> the value of t statistic is,

t = 0,

df = 3

p-value = 0.5

Decision Rule: If p-value greater than 0.05 level of significance then we accept the null hypothesis.
From above results p-value(0.5) > 0.05
So we accept the null hypothesis here

i.e. The average cost of the first design is higher than the second.

The sample size at 10% level of significance is,

sample size = [ Z^2 * p * (1-p)] /e^2

Z is critical value at 10% level of significance 1.2816

p = 0.06

e = margin of error 0.01

sample size = 154.3949

i.e. sample size = 155

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