7. During crash tests, the National Highway Traffic Safety Administration record

Question

7. During crash tests, the National Highway Traffic Safety Administration records the severity of a driver’s head injury when the car is in a head-on collision with a fixed barrier while traveling at 35 miles per hour. The more points assigned, the more severe the injury. The ratings are normally distributed with a mean of 605 and a standard deviation of 185.

a. What is the probability that the rating will fall between 500 and 700?

b. What is the probability that the rating will fall between 400 and 500?

c. What is the probability that the rating will be less than 850?

d. What is the probability that the rating will be greater than 1000?

e. What rating will only 10% of the crash-tested cars exceed?

Explanation / Answer

a)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    500      
x2 = upper bound =    700      
u = mean =    605      
          
s = standard deviation =    185      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.567567568      
z2 = upper z score = (x2 - u) / s =    0.513513514      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.285164317      
P(z < z2) =    0.69620392      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.411039602   [ANSWER]

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b)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    400      
x2 = upper bound =    500      
u = mean =    605      
          
s = standard deviation =    185      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1.108108108      
z2 = upper z score = (x2 - u) / s =    -0.567567568      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.133907565      
P(z < z2) =    0.285164317      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.151256752   [ANSWER]

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c)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    850      
u = mean =    605      
          
s = standard deviation =    185      
          
Thus,          
          
z = (x - u) / s =    1.324324324      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   1.324324324   ) =    0.907302322 [ANSWER]

******************

d)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    1000      
u = mean =    605      
          
s = standard deviation =    185      
          
Thus,          
          
z = (x - u) / s =    2.135135135      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   2.135135135   ) =    0.016374987 [ANSWER]

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e)

First, we get the z score from the given left tailed area. As          
          
Left tailed area = 1 - 0.10 =    0.9      
          
Then, using table or technology,          
          
z =    1.281551566      
          
As x = u + z * s,          
          
where          
          
u = mean =    605      
z = the critical z score =    1.281551566      
s = standard deviation =    185      
          
Then          
          
x = critical value =    842.0870396   [ANSWER]  

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