A new design for the braking system on a certain type ofcar has been proposed. For the current system, the trueaverage braking distance at 40 mph under specified con-ditions is known to be 120 ft. It is proposed that the newdesign be implemented only if sample data stronglyindicates a reduction in true average braking distance forthe new design.a. Define the parameter of interest and state the relevanthypotheses.b. Suppose braking distance for the new system isnormally distributed with s 5 10. Let X denote thesample average braking distance for a randomsample of 36 observations. Which values of x aremore contradictory to H0 than 117.2, what is theP-value in this case, and what conclusion is appro-priate if a 5 .10?

a. The parameter of interest in this case is the true average braking distance for the new design of the braking system. The relevant hypotheses are:

Null hypothesis (H0): The true average braking distance for the new design is equal to or greater than the current braking distance of 120 ft. Alternative hypothesis (Ha): The true average braking distance for the new design is less than 120 ft.

b. Given that the braking distance for the new system is normally distributed with a standard deviation (s) of 10, and X represents the sample average braking distance for a random sample of 36 observations, we need to determine which values of x are more contradictory to H0 than 117.2.

To do this, we can calculate the z-score for 117.2 using the formula:

z = (x - μ) / (s / √n)

where x is the observed sample average braking distance, μ is the hypothesized true average braking distance under H0 (120 ft), s is the standard deviation, and n is the sample size (36).

In this case, the z-score is:

z = (117.2 - 120) / (10 / √36) = -0.8

To find the P-value, we need to determine the probability of observing a z-score less than or equal to -0.8. We can consult a standard normal distribution table or use statistical software to find this probability. Assuming a significance level (α) of 0.10, the P-value is the probability of observing a z-score less than or equal to -0.8.

If the P-value is less than or equal to α (0.10), we reject the null hypothesis. If the P-value is greater than α, we fail to reject the null hypothesis.

The conclusion in this case would be based on the P-value and the chosen significance level.