Question

A certain model of automobile has its gas mileage (in miles per gallon, or mpg) normally distributed, with a mean of 26 mpg and a standard deviation of 4 mpg. Find the probability that a car selected at random has the following gas mileages. (Round your answers to four decimal places.)

(a) less than 20 mpg
(b) greater than 28 mpg
(c) between 24 and 28 mpg

Answer 1

het X denote the gas mileage of automobile. Given gas mileage of automobile is normally distributed with mean of 26 mpg and a Standard deviation of 4mpg. Then X is a normal variate with parameters M= = 26 mpg and = 4 mpg Then the standard normal wariate Za X-A r

X-26 a) that a Can selected at random [*=240 Z 20-26 4 Probability has mileage less than 20 mpg PLX x < 20) EP [ pla 67 - p[z <^3/2] - P[72-165] =P[-«o<z20] – P[152240 - P [ 4 Since the standard normal came is Symmetrie about the Ordinate at Z= p [- <z2o] D area from o to

NU 0.5 P [-1.52720) area from o to 1.5 0.4332 Then P(x 2 го) p [-oczLo] - pEigaza? 1] 0.5 -0.4332 = O 06 68

Z

Then the probability that a car selected at random has mileage less than 20mpg = 0.668

b)

b) Porobability that a Car Selected at randons has mileage greater than 28 mpg = PC x >28) =P (x726 > 20:26 P [z>$) ap[2> +) P[z>015) P[022os] - P[0 < 22013]

From the table of standard normal curve, we get  $P \left [ 0<Z<0.5 \right ]=0.1915$

Then

$P\left [ X>28 \right ]=P\left [0<Z<\infty \right ]-P \left [ 0<Z<0.5 \right ]$

$=0.5-0.1915$

$=0.3085$

The the probability that a car selected at random has mileage grater than 28mpg = 0.3085

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0 015 元

c)

c ) probability that a Car Selected at random has mileage between 24 and 28 mpg P ( 242 x 228) P

P 24-26 LE < X-26 28-20 Z Z < za ***** PC p(0.522 P(-0.522 <o] + P[ocz2015] 2 1 <z20.5 15 11 Since the standaard normal lune is Symmetric about the ordinate 2=0 p[1015Lz2o Area from o to 0.5 } 0.1915 Then L 28 P (24.2% 28) P[-0.52220] + P[02220.5] 3

(Area from to 0.5 +(Area + ( Area from 0 to 0.5 u O. 1915 ton 1915 0383

7 0.5 وہ۔

Then the probability that a car selected at random has mileage between 24mpg and 28mpg = 0.383