Question

For each of the following two-samples t-tests (problems 1-6): (a) Determine if a F test for the ratio of two variances is appropriate to calculate for the context. If it is appropriate, conduct the analysis and report the result. Include what statistical conclusion you should draw from the analysis (i.e., whether you should conduct a pooled-variance t-test or an unequal-variances t-test). (b) Identify the most appropriate t-test to conduct for the situation/data given. Don’t forget to consider if the context requires one/two-tail tests. (c) Provide a statistical and practical interpretation of your findings.

5. An important feature of digital cameras is battery life—the number of shots that can be taken before the battery needs to be recharged. The file Cameras contains the battery life information of 11 subcompact cameras and 7 compact cameras (Data extracted from “Cameras,” Consumer Reports, July 2012, pp. 42-44). Is there evidence of subcompact cameras have better battery life than compact cameras? (Use a 0.05 level of significance)

Subcompact	Compact
140	230
200	220
190	160
240	240
160	200
240	300
190	120
200
190
210
210

Answer 1

For Subcompact :

∑x = 2170

∑x² = 436900

n1 = 11

Mean , x̅1 = Ʃx/n = 2170/11 = 197.2727

Standard deviation, s2 = √[(Ʃx² - (Ʃx)²/n)/(n-1)] = √[(436900-(2170)²/11)/(11-1)] = 29.6954

For Compact :

∑x = 1470

∑x² = 328900

n2 = 7

Mean , x̅2 = Ʃx/n = 1470/7 = 210.0000

Standard deviation, s2 = √[(Ʃx² - (Ʃx)²/n)/(n-1)] = √[(328900-(1470)²/7)/(7-1)] = 58.0230

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a)

Null and alternative hypothesis:

Hₒ : σ₁ = σ₂ ; H₁ : σ₁ ≠ σ₂

Test statistic:

F = s₂² / s₁² = 58.023² / 29.6954² = 3.8179

Degree of freedom:

df₁ = n₂-1 = 6

df₂ = n₁-1 = 10

P-value = 2*F.DIST.RT(3.8179, 6, 10) = 0.0609

Conclusion:

As p-value > α, we fail to reject the null hypothesis.

We should conduct a pooled-variance t-test.

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b)

Null and Alternative hypothesis:

Ho : µ1 ≤ µ2

H1 : µ1 > µ2

Pooled variance :

S²p = ((n1-1)*s1² + (n2-1)*s2² )/(n1+n2-2) = ((11-1)*29.6954² + (7-1)*58.023²) / (11+7-2) = 1813.6364

Test statistic:

t = (x̅1 - x̅2) / √(s²p(1/n1 + 1/n2 ) = (197.2727 - 210) / √(1813.6364*(1/11 + 1/7)) = -0.6181

df = n1+n2-2 = 16

p-value = T.DIST.RT(-0.6181, 16) = 0.7274

Decision:

p-value > α, Do not reject the null hypothesis

Conclusion:

There is not enough evidence to conclude that subcompact cameras have better battery life than compact cameras.