A new design for the braking system on a certain type of car

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 c strongly indicates a reduction in true average braking distance for the new design. Define the parameter of interest. p = true average braking distance for the old design p = true average braking distance for the new design p = true proportion of cars whose braking distances did not reduce p = true proportion of cars whose braking distances reduced State the relevant hypotheses. What is the P-value in this case? (Round your answer to four decimal places.) Suppose braking distance for the new system is normally distributed with a = 11. Let X denote the sample average braking distance for a random sample of 36 observations. Which values of x are more contradictory to H_0 than 117.2? What conclusion is appropriate if a = 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. 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.)

Solution

As the concern really is the average braking distance, then

a)

u = true average braking distance for the new design. [ANSWER, B]

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Ho: u = 120
Ha: u < 120 [ANSWER]

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b)

Ha is contradictory to Ho, so we use <,

Xbar <= 117.2 [ANSWER]

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As we can see, this is a    left   tailed test.      
              
              
Getting the test statistic, as              
              
X = sample mean =    117.2          
uo = hypothesized mean =    120          
n = sample size =    36          
s = standard deviation =    11          
              
Thus, z = (X - uo) * sqrt(n) / s =    -1.527272727          
              
Also, the p value is              
              
p =    0.063346604   [ANSWER, p value]

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As P < 0.10, we   REJECT THE NULL HYPOTHESIS.      

Thus, OPTION A: The new design does have a mean breaking distance less than 120 ft at 40 mph. [ANSWER, A]

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c)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    120      
u = mean =    115      
n = sample size =    36      
s = standard deviation =    11      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    2.727272727      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   2.727272727   ) =    0.003193012 [ANSWER]