Q) A new design for the braking system on a certain type of car has been propose

Question

Q) A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120 ft. It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design.

A) Download the dataset “brakingdistance.csv” and carry out a hypothesis test (clearly write down each of the 4 steps for the hypothesis test) to decide whether the new design should be implemented. Use significance level of 0.05(8pts)

B) Verify the test you use is appropriate. For example, if you use a t-test or z-test, verify the corresponding conditions. (2pts)

Use Rcommander if possible.

Dataset “brakingdistance.csv”:

Explanation / Answer

Solution

Back-up Theory

Let X = breaking distance (in feet). It is reasonable to assume X ~ N(µ, 2), where µ = mean = standard deviation of the population.

To test if the new design brings in a reduction in true average braking distance, as compared to the existing average braking distance of 120 feet,

we set

Null Hypothesis. H0: µ = µ0 = 120    Vs Alternative, HA: µ < µ0

Test Statistic for this test is: Z = (n)(Xbar - µ0)/ if is known ……………..(1)    

t = (n)(Xbar - µ0)/s if is unknown……………(2)

where n = sample size, Xbar = sample average and s = sample standard deviation.

Under H0, Z ~ N(0, 1) and t ~ tn – 1 ……………………………………………..(3)

Decision Criterion: Reject H0, at level of significance %, if calculated value of the appropriate test statistic is less than the lower % point of the respective distribution.

[note that lower % point is to be used since the alternative is one-sided less than type]

Now, to work out the solution,

Given n = 30, = 0.05 (5%). Since is not given, we will use its estimate, s and hence t-statistic.

Part (a)

From the given data, using Excel Function, the results of calculations are summarized below:

Xbar = 114.28; s = 3.82; t = - 8.191

From Standard Statistical Tables, lower t29, 0.05 = - 1.699

Inference: Since calculated value of t is less than the lower 5% point of t29, H0 is rejected.

Conclusion: There is sufficient evidence to suggest that the new design brings in a reduction in true average braking distance   

DONE

Part (b)

As already indicated, our use of t-test is appropriate since is not given. DONE

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