Respuesta :
Answer:
A)[tex]X \sim N(68 , 25)[/tex]
B) the probability that is traveling more than 70 mph is 0.3446
C) the probability that it is traveling between 65 and 75 mph is 0.6449
D) 90% of all cars travel at least 56.85 mph fast on the freeway
Step-by-step explanation:
The speed at which cars go on the freeway is normally distributed with mean 68 mph and standard deviation 5 miles per hour.
Mean = [tex]\mu = 68 mph[/tex]
Standard deviation = [tex]\sigma = 5 mph[/tex]
A) X ~ N( _____, _______ )
In general [tex]X \sim N( \mu , \sigma^2)[/tex]
[tex]\mu = 68 mph[/tex]
[tex]\sigma = 5 mph[/tex]
[tex]\sigma^2 = 5^2 = 25[/tex]
So, [tex]X \sim N(68 , 25)[/tex]
B) If one car is randomly chosen, find the probability that is traveling more than 70 mph.i.e.P(X>70)
So,[tex]Z = \frac{x-\mu}{\sigma}\\Z=\frac{70-68}{5}[/tex]
Z=0.4
Using Z table
P(Z>70)=1-P(Z<70)=1-0.6554=0.3446
Hence the probability that is traveling more than 70 mph is 0.3446
C) If one of the cars is randomly chosen, find the probability that it is traveling between 65 and 75 mph.
P(65<X<75)
[tex]Z = \frac{x-\mu}{\sigma}[/tex]
AT x = 65
[tex]Z=\frac{65-68}{5}[/tex]
Z=-0.6
AT x = 75
[tex]Z=\frac{75-68}{5}[/tex]
Z=1.4
Using Z table
P(65<X<75)=P(-0.6<Z<1.4)=P(Z<1.4)-P(Z<-0.6)=0.9192-0.2743=0.6449
Hence the probability that it is traveling between 65 and 75 mph is 0.6449
D)90% of all cars travel at least how fast on the freeway?
Since we are supposed to find at least how fast on the freeway
So,P(X>x)=0.9
1-P(X<x)=0.9
1-0.9=P(X<x)
0.1=P(X<x)
Z value at 10% =-2.23
So, [tex]Z=\frac{x-\mu}{\sigma}\\-2.23=\frac{x-68}{5}\\-2.23 \times 5 =x-68\\(-2.23 \times 5)+68=x[/tex]
56.85 = x
Hence 90% of all cars travel at least 56.85 mph fast on the freeway
