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Suppose you are an expert on the fashion industry and wish to gather information to compare the amount earned per month by models featuring Liz Claiborne attire with those of Calvin Klein. The following is the amount ($000) earned per month by a sample of 15 Claiborne models:

$4.7 $3.8 $6.7 $3.8 $3.9 $4.2 $4.0 $4.0 $4.8 $3.1 3.1 5.5 3.4 6.1 4.9

The following is the amount ($000) earned by a sample of 12 Klein models.

$4.0 $3.8 $3.6 $2.5 $4.9 $5.2 $5.9 $4.9 $4.4 $3.6 5.1 4.7

Required:
a. Find the degrees of freedom for unequal variance test.
b. State the decision rule for 0.05 significance level: H0: μLC ≤ μCK; H1: μLC > μCK.
c. Compute the value of the test statistic.
d. Is it reasonable to conclude that Claiborne models earn more? Use the 0.05 significance level.

Sagot :

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Answer:

24 ; Tstat > Tcritical ; 0.052 ; Fail to eject the Null

Step-by-step explanation:

Claiborne model:

$4.7 $3.8 $6.7 $3.8 $3.9 $4.2 $4.0 $4.0 $4.8 $3.1 $3.1 $5.5 $3.4 $6.1 $4.9

Sample size, n1 = 15

Using calculator :

Mean, m1 = 4.4

Standard deviation, s1 = 1.056

Klein model:

$4.0 $3.8 $3.6 $2.5 $4.9 $5.2 $5.9 $4.9 $4.4 $3.6 $5.1 $4.7

Sample size, n = 12

Mean, m2 = 4.38

Standard deviation (s) = 0.923

To obtain the degree of freedom for unequal variances test, use the Relation :

Degree of freedom (DF) = (s1²/n1 + s2²/n2)² / { [ (s1² / n1)² / (n1 - 1) ] + [ (s2² / n2)² / (n2 - 1) ] }

(1.056^2/15 + 0.923^2/12)^2 / {[(1.056^2/15)^2 / (15 - 1)] + [(0.923^2/12)^2 / (12 - 1)]}

0.0211226 / (0.0003947 + 0.0004581)

= 24.768527

df = 24 rounded down to the nearest whole number.

Decision region :

α = 1-0.95 = 0.05

Tdf, 0.05 = 1.711

Tcritical = 1.711

Reject Null, if Tstat > Tcritical

The t - test statistic :

(m1 - m2) / (s1²/n1 + s2²/n2)

(4.4 - 4.38) / sqrt(1.056^2 / 15 + 0.923^2 / 12)

0.02 / 0.3812301

= 0.0524617

= 0.052

Since, Tstat < Tcritical, we fail to reject the Null. It is not reasonable to conclude that Claiborne model earns more.

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