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.

(a) Define the parameter of interest.

μ = true average braking distance for the new design μ = true average braking distance for the old design     p̂ = true proportion of cars whose braking distances reduced p̂ = true proportion of cars whose braking distances did not reduce

State the relevant hypotheses.

H₀: p̂ = 120
H_a: p̂ < 120 H₀: μ = 120
H_a: μ ≠ 120     H₀: μ = 120
H_a: μ > 120 H₀: μ = 120
H_a: μ < 120 H₀: p̂ = 120
H_a: p̂ ≠ 120

(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?

x ≥ 117.2 x ≤ 117.2    

What is the P-value in this case? (Round your answer to four decimal places.)


What conclusion is appropriate if α = 0.10?

The new design does have a mean breaking distance less than 120 feet at 40 mph. The new design does not have a mean breaking distance less than 120 feet at 40 mph.    

(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.)

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

Answer:-

Given That:-

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) Define the parameter of interest.

= true average braking for the new design.

Null hypothesis, H₀ :   = 120

Alternative hypothesis, H₁ : < 120

(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?

From the given information, 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

The correct answer is:  

Calculate the test statistic value.

  

Z = -1.52727

Calculate the P - value.

P - Value = P (Z < -1.52727)

= (=NORMSDIST(-1.52727)) (Use MS Excel)

= 0.06334694

= 0.0633 (Round to 4 decimal place)

The level of significance is :  

Decision: Reject H₀, because, the P - value is less than the 0.10 level of significance.

Conclusion: The new design does have a mean breaking less than 120 feet at 40 mph.

(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.)

Calculate the probability that the new design is not implemented when its true average braking distance is actually 115 ft.

= P(Z > 1.2)

= 1 - (=NORMSDIST(1.2)) (Use MS Excel)

= 1 - 0.8849

= 0.1151

Hence,

The required probability is : 0.1151

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