Question

A marketing consultant was hired to visit a random sample of five sporting goods stores across the state of California. Each store was part of a large franchise of sporting goods stores. The consultant taught the managers of each store better ways to advertise and display their goods. The net sales for 1 month before and 1 month after the consultant's visit were recorded as follows for each store (in thousands of dollars):

Store	1	2	3	4	5
Before visit	57.3	94.4	49.2	77.4	43.2
After visit	63.7	101.8	57.8	81.2	41.9

Do the data indicate that the average net sales improved? Use α = 0.05. (Let d = before − after.)

What are we testing in this problem?

paired differencesingle mean    difference of meanssingle proportiondifference of proportions

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ_d > 0; H₁: μ_d = 0H₀: μ_d = 0; H₁: μ_d < 0    H₀: μ_d = 0; H₁: μ_d > 0H₀: μ_d = 0; H₁: μ_d ≠ 0

(b) What sampling distribution will you use? What assumptions are you making?

The Student's t. We assume that d has an approximately normal distribution.The standard normal. We assume that d has an approximately normal distribution.    The Student's t. We assume that d has an approximately uniform distribution.The standard normal. We assume that d has an approximately uniform distribution.

What is the value of the sample test statistic? (Round your answer to three decimal places.)


(c) Find (or estimate) the P-value.

P-value > 0.2500.125 < P-value < 0.250    0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.05 to conclude that the average net sales have improved.There is insufficient evidence at the 0.05 to conclude that the average net sales have improved.    

Answer 1

paired difference

a_)

level of significance=0.05

H₀: μ_d = 0; H₁: μ_d < 0

The Student's t. We assume that d has an approximately normal distribution.

mean dbar=	d̅     =	-4.980
degree of freedom =n-1                            =		4.000
Std deviaiton SD=√(Σd2-(Σd)2/n)/(n-1) =		3.930903
std error=Se=SD/√n=		1.7580
test statistic            =	(d̅-μd)/Se         =	-2.833

0.005 < P-value < 0.025

d)

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant
e)

There is sufficient evidence at the 0.05 to conclude that the average net sales have improved.