Question

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

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.

(b) Suppose braking distance for the new system is normally distributed with σ = 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 conclusion is appropriate if α = 0.10?

(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.)

(b)

Test is left tailed. The rejection region is

$R=\left \{ \bar{x}\leq 117.2 \right \}$

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Decision: Since p-value is less than 0.10 so we reject the null hypothesis.

(c)

The design will not be implemented if we fail to reject the null hypothesis.

Since test is left tailed so The critical value of z for which we will reject the null hypothesis for 0.10 level of significance is -1.28.

$z=\frac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\frac{\bar{x}_{c}-120}{11/\sqrt{36}}$

$-1.28=\frac{\bar{x}_{c}-120}{11/\sqrt{36}}$

$\bar{x}_{c}=117.65$

The z-score for $\bar{x}_{c}=117.65$ and $\mu=115$ is

$z=\frac{117.65-115}{11/\sqrt{36}}=1.45$

The type II error is

$\beta=P(z>1.45)=1-P(z\leq 1.45)=0.0735$