Question

11-4

A marketing analyst is studying the variability in customer purchase amounts between shopping mall stores and “big box” discount stores. She suspects the variability is different between those stores due to the nature of the customers involved. To investigate this issue in detail, she compiled two random samples, each consisting of 26 purchase amounts at shopping mall stores (sample 1) and discount stores (sample 2).

Shopping Mall Store Purchase	Discount Store Purchase
226	133
49	96
245	75
346	38
60	115
106	117
110	145
95	110
85	172
110	39
234	84
319	75
76	127
114	66
114	173
98	135
143	92
104	33
229	129
232	87
114	128
198	125
146	111
156	95
213	94
196	92

a. Select the hypotheses to test whether the variance of the purchase amounts differs between the two types of stores.

• H₀: σ1² / σ2² = 1, H_A: σ1² / σ2² ≠ 1

• H₀: σ1² / σ2² ≤ 1, H_A: σ1² / σ2² > 1

• H₀: σ1² / σ2² ≥ 1, H_A: σ1² / σ2² < 1

b. Construct the 90% confidence interval for the ratio of the population variances. Assume that purchase amounts are normally distributed. (Round "F" value and final answers to 2 decimal places.)

c. Use the confidence interval to test whether the variance of the purchase amounts differs between the two stores at the 10% significance level.

The 90 % confidence interval (contains, does not contain) the value 1. we (reject, do not reject) HO and conclude that the population variances in purchase amounts are (different, not different) between shopping mall stores and "bid box" stores.

d-1. Find the p-value.

• p-value < 0.01

• p-value  0.10
• 0.05  p-value < 0.10
• 0.02  p-value < 0.05
• 0.01  p-value < 0.02

d-2. Confirm your conclusion by finding the p-value.

Thus, our conclusion (agrees, do agree) with the confidence interval in part c.

Answer 1

(a) Here hypothesis is

H₀ : σ₁² / σ₂² = 1

H_a : σ₁² / σ₂² ≠ 1

(b) Here sample standard deviatiom for population 1 = s₁ = 78.3153

sample standard deviatiom for population 2 = s₂ = 36.5358

so here, n1= n2= 26

Lower Limit = s₁²/s₂² * 1/[^F_]

^F = FINV(0.10, 25, 25) = 1.6831

Lower Limit = (78.3153/36.5358)² * (1/1.6831) = 2.73

Upper Limit = s₁²/s₂² * [^F_]

^F = FINV(0.10, 25, 25) = 1.6831

Lower Limit = (78.3153/36.5358)² *1.6831 = 7.73

90% confidence interval = (2.73, 7.73)

(c) The 90 % confidence interval contains the value 1. we reject H_O and conclude that the population variances in purchase amounts are different between shopping mall stores and "bid box" stores.

(d) Here p value = FDIST(4.595, 25,25) = 0.00015

so p-value < 0.01

(e) Thus, our conclusion agrees with the confidence interval in part c.