Question

If you​ research, the average time in which​ driver-free cars will be used in the general population is around 20 years with a standard deviation of 10 years. Assume that the time is normally distributed.

1. What is the probability that a single randomly selected person believes that the time is between 15 and 20​ years? (round to four​ decimals) nothing

2. What is the probability that a sample of 15 people believes that the average time is between 15 an 20​ years? (Round to four​ decimals) nothing

Answer 1

Given,

= 20 , = 10

We convert this to standard normal as

P( X < x) = P( Z < x - / )

a)

P( 15 < X < 20) = P (X < 20) - P( X < 15)

= P( Z < 20 - 20 / 10) - P( Z < 15 - 20 / 10)

= P( Z < 0) - P( Z < -0.5)

= 0.5 - 0.3085 (Probability calculated from Z table)

= 0.1915

b)

Using central limit theorem,

P( < x) = P (Z < x - / ( / sqrt(n) ) )

So,

P(15 < < 20) = P( < 20) - P( < 15)

= P( Z < 20 - 20 / ( 10 / sqrt(15) ) ) - P( Z < 15 - 20 / ( 10 / sqrt(15) ) )

= P( Z < 0) - P( Z < -1.9365)

= 0.5 - 0.0264 (Probability calculated from Z table)

= 0.4736