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 for the new design. Define the parameter of interest and state the relevant hypothesis. Suppose braking distance for the new system is normally distributed with sigma = 10. Let X denote the sample average braking distance for a random sample of 36 observations. Which of the following three rejection regions is appropriate : What is the significance level for the appropriate region of part(b)?How would you change the region to obtain a test with alpha = 0.001? What is the probability that the new design is not implemented when its true average braking distance is actually 115 ft and the appropriate region from part (b) is used? Let Z = (x - 120)/ (sigma / n), what is the significance level for the region {z : z le -2.33}? For the region {z : z le -2.88}?

Answer 1

Answer:

Step-by-step explanation:

Hello!

a. The study variable is X: "breaking distance using the new design"(ft)

X~N(μ;σ²)

The parameter of interest is μ: The population mean of the breaking distance using the new design.

The hypothesis states that the new design reduces the population mean of the braking distance. Symbolically:

H₀: μ≥120

H₁: μ<120

Unfortunately, you didn't copy the choices for the critical regions. With this type of hypothesis, you have a one-tailed critical region and will reject the null hypothesis to small values of the statistic. I've looked for the critical regions using the most common significance levels and the one in part b.

Significance level α: 10%

Critical value $Z_{\alpha } = Z_{0.10} = -1.28$

Significance level α: 5%

$Z_{\alpha } = Z_{0.05} = -1.64$

Significance level α: 1%

$Z_{\alpha } = Z_{0.01} = -2.33$

b. Significance level α: 0.1%

$Z_{\alpha } = Z_{0.001} = -3.107$

The significance level is stated by the investigator before making the test, what changes is the rejection region, with a lower level of significance you get a smaller rejection region and is more likely to support a false null hypothesis. On the other hand, the smaller the level of significance, the bigger the power of the test so if you end up rejecting the null hypothesis you will have more certainty of having made a correct decision.

c. The null hypothesis states that the population mean of the braking distance is 120ft or more and you conduct the test supposing this hypothesis is true, that's why you can use the number in this hypothesis when calculating the statistic value.

If later it is found out that the population mean is 115 ft, that means your null hypothesis is FALSE.

And if the condition for implementing the new system is that the braking distance is reduced, since it wasn't applied you can assume that the null hypothesis WASN'T REJECTED.

So, in this case, it is a Type II error situation, You didn't reject the null hypothesis but it was false and its associated probability is β.

d.

{z : z le -2.33} significance level is 0.0099 ≅ 0.01

{z : z le -2.88} significance level is 0.00199 ≅ 0.02

I hope it helps!