A new design for the braking system on a certain type of car has been proposed.

Question

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. Suppose braking distance for the new system is normally distributed with = 11. Let

X

denote the sample average braking distance for a random sample of 36 observations. Which values of

x

are more contradictory to H0 than 117.2?

(b) Suppose braking distance for the new system is normally distributed with = 11. Let

X

denote the sample average braking distance for a random sample of 36 observations. Which values of

x

are more contradictory to H0 than 117.2?

What is the P-value in this case?

(c) What is the probability that the new design is not implemented when its true average braking distance is actually 115 ft and the test from part (b) is used? (Round your answer to four decimal places.)

Explanation / Answer

Solution:

Part b

Here, we have to use one sample z test for population mean.

H0: µ = 120 versus Ha: µ < 120

Test statistic = Z = (Xbar - µ) / [/sqrt(n)]

We are given

Sample size = n = 36

Sample mean = Xbar = 117.2

Population mean = µ = 120

Population standard deviation = = 11

Now, plug all values in the above formula

Z = (117.2 – 120) / [11/sqrt(36)]

Z = -1.5273

P-value = 0.0633

(By using z-table)

Part c

Here, we are given

Sample size = n = 36

Sample mean = Xbar = 115

Population mean = µ = 120

Population standard deviation = = 11

Test statistic = Z = (Xbar - µ) / [/sqrt(n)]

Z = (115 – 120) / [11/sqrt(36)]

Z = -2.7273

P(Z<-2.7273) = 0.0032

Required probability = 0.0032

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