A simple random sample of 13 four-cylinder cars is obtained, and the braking distances are measured. The mean braking distance is 137.5 ft and the standard deviation is 5.8 ft. A simple random sample of 12 six-cylinder cars is obtained, and the braking distances have a mean of 136.3 ft with a standard deviation is 9.7 ft. Use a 0.05 significance level to test thee claim that the mean braking distance of four-cylinder cars is greater than the mean braking distance of six-cylinder cars.
(a) CLAIM: IN WORDs
( b) CLAIM: IN EQUATION FORM
(c) DETERMINE THE HYPOTHESES.
(d) DRAW THE DISTRIBUTION -LABEL THE CRITICAL VALUE(S).
( e) CALCULATE THE TEST STATISTIC and include the test statistic on the graph above.
(f)STATE YOUR CONCLUSION IN CONTEXT OF H g.
(g)STATE YOUR CONCLUSION IN CONTEXT OF THE CLAIM.

F-distribution is "probability density function especially is used in analysis of variance and is function of ratio of two independent random variable each of which has chi-square distribution and its divided by number of degrees of freedom".

According to the question,

sample size n₁ =13 four cylinder cars

sample size n₂ = 12 six cylinder cars

mean of four cylinder cars μ= 137.5ft

mean of six cylinder cars μ= 136.3ft

standard deviation S₁² = 5.8ft

standard deviation S₂² = 9.7ft

H₀:S₁² > S₂² (The mean braking distance of four-cylinder cars is greater than the mean braking distance of six cylinder cars)

H₁:S₁²≠S₂² (The mean braking distance of four-cylinder cars is not greater than the mean braking distance of six cylinder cars)

Formula for F-distribution = S₁²/S₂²

= 5.8/9.7 =0.59793

Formula to calculate number of degrees of freedom for F-distribution (n₁-1,n₂-2).

= (13-1,12-2)

=(12,10)

The significant values of F - distribution for left tailed test at 0.05 level of significance. Tabulated F₍₁₂,₁₀₎ values lies between 2.76 to 4.30. Since, calculated values F-distribution is less than the tabulated value F₍₁₂,₁₀₎. H₀ may be accepted.

Hence, The mean braking distance of four-cylinder cars is greater than the mean braking distance of six cylinder cars.

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