For a particular model of car, the miles per gallon (mpg) rating obtained for hi

Question

For a particular model of car, the miles per gallon (mpg) rating obtained for highway driving follows a Normal distribution with mean 43.7 mpg and standard deviation 2.3 mpg. Use this information to answer the following questions.
(a) What is the probability a car will obtain a miles per gallon rating for highway driving greater than 40 mpg? Answer this question by completing parts (i) and (ii). i. Provide the z-score corresponding to 40 mpg. ii. Based on your answer in (i), what is the probability a car will obtain a miles per gallon rating for highway driving greater than 40 mpg?
(b) What is the probability a car will obtain a miles per gallon for highway driving less than 45 mpg? Answer this question by completing parts (i) and (ii). i. Provide the z-score corresponding to 45 mpg. ii. Based on your answer in (i), what is the probability a car will obtain a miles per gallon for highway driving less than 45 mpg?
(c) Find Q1 (First Quartile). Answer this question by completing parts (i) and (ii). i. Provide the z-score corresponding to Q1. ii. Based on your answer in (i), provide the value of Q1.
(d) Find Q3 (Third Quartile). Answer this question by completing parts (i) and (ii). i. Provide the z-score corresponding to Q3. ii. Based on your answer in (i), provide the value of Q3.
(e) What is the value of the IQR for the distribution of miles per gallon ratings for highway driving?
(f) What is the mpg rating such that only 14% of all cars have an mpg rating larger than that mpg value? Answer this question by completing parts (i) and (ii). i. Provide the z-score corresponding to the above statement. ii. Based on your answer in (i), what is the mpg rating such that only 14% of all cars have an mpg rating larger than that mpg value? 2 iii. What percentile does the mpg rating obtained in part (ii) correspond to? (Enter value as a whole number, not a proportion.)
(g) What proportion of cars will have an mpg rating within 3 mpg of the mean? Answer this question by completing parts (i) through (iii). i. Provide the SMALLER z-score corresponding to the above statement. ii. Provide the LARGER z-score corresponding to the above statement. iii. Based on your answers in (i) and (ii), what proportion of cars will have an mpg rating within 3 mpg of the mean?

Explanation / Answer

a)(i)for car speed 40mph ; zscore =(40-43.7)/2.3 =-1.6087

(ii) P(Z>-1.6087) =1-P(Z<-1.6087|) =1-0.0538 =0.9462

b) (i)

for car speed 45mph ; zscore =(45-43.7)/2.3 =0.5652

(ii) P(Z<0.5652) =0.7140

c) (i) for 25 th percentile =-0.6745

(ii) hence Q1 scor =43.7-0.6745*2.3=42.15

d)for Q3; 75th percentile z score =0.6745

hence Q3 score =43.7+0.6745*2.3=45.25

e)IQR =Q3-Q1 =45.25-42.15 =3.10

f)fopr 14% larger, value of z at 86 percentile 1.0803

hence corresponding value =43.7+1.0803*2.3 =46.18

g)for 3 mpg below zscore =(-3/2.3) =-1.3043

for 3 mpg above zscore =1.3043

hence P(-1.3043<Z<1.3043) =0.9039-0.0691=0.8078

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