Question

The age of professional racecar drivers is normally distributed with a mean of 30 and a standard deviation of 6. One driver is randomly selected. (Round all answers to at least three decimal places)

What is the probability that this driver is less than 20 years old?





What is the probability that this driver is more than 12 years old?





What is the probability that this driver is between 31 and 35 years old?





95% of all drivers are younger than what?
(Hint : 95% of all values of Z are below it, and 5% are above it. Use the Z-table to find the Z value and then use the formula x=(Z⋅σ)+μx=(Z⋅σ)+μ to find the xx value.)

Answer 1

Solution :

Given that ,

mean = = 30

standard deviation = = 6

P(x < 20 ) = P[(x - ) / < ( 20 - 30) / 6 ]

= P(z < -1.67 )

Using z table,

= 0.0475

Probability = 0.0475

b.

P(x > 12 ) = 1 - P( x < 12 )

= 1- P[(x - ) / < ( 12 - 30 ) / 6 ]

=1- P(z < -3 )

Using z table,

= 1 - 0.0013

= 0.9987

Probability = 0.9987

c.

P( 31 < x < 35 )

= P[( 31 - 30 ) / 6 ) < (x - ) /  < ( 35 - 31) / 6 ) ]

= P( 0.17 < z < 0.83 )

= P(z < 0.83 ) - P(z < 0.17)

Using z table,

= 0.7967 - 0.5675

= 0.2292

Probability = 0.2292

d.

The z - distribution of the 95% is,

P(Z > z) = 95%

= 1 - P(Z < z ) = 0.95

= P(Z < z ) = 1 - 0 .95

= P(Z < z ) = 0.05

= P(Z < -1.645 ) = 0.05  

z = -1.645

Using z-score formula,

x = z * +

x = -1.645 * 6 + 30

x = 20.13

Answer : x = 20.13