Question

The previous question concerning drive times from Rexburg and CEI used too simplistic a model. In actuality, the drive times are normally distributed with mean drive time of μ=32μ=32 minutes and a standard deviation of σ=5.σ=5.

What is the probability that a drive will take less than 30.9 minutes? (Use two decimal places)


Using the empirical rule, 68% of all drive times should fall within what interval?


What drive times would be considered unlikely (or highly unlikely) to happen? (Select all correct answers)

• Less than 18 minutes
• Less than 23 minutes
• Less than 27 minutes
• More than 37 minutes
• More than 44 minutes
• More than 49 minutes

The 4% longest drives require at least how many minutes? (round to 1 decimal place)

Answer 1

Drive times are normally distributed with mean drive time of 32 minutes and standard deviation of 5 minutes.

Let, X be the random variable denoting the drive time.

Then, X follows normal with mean 32 and standard deviation of 5.

So, we can say that

Z=(X-32)/5 follows standard normal with mean 0 and standard deviation of 1.

Question 1)

P(Drive time will take less than 30.9 minutes)

Where, phi is the distribution function of the standard normal variate.

From the standard normal table, this becomes

The answer, rounded off to two decimal places, is 0.41.

Question 2)

We note that within one standard deviations of the mean, 68% of the population lies.

Now, within one one standard deviation, implies the interval (Mean-Standard deviation, Mean+Standard deviation).

Here,

Mean-Standard deviation=32-5=27

Mean+Standard deviation=32+5=37

The answer is

68% of the drive times should fall in the interval 27 minutes to 37 minutes.

Question 3)

Formula for z-score is

Drive time less than 18 minutes

Drive time less than 23 minutes

Drive time less than 27 minutes

Drive time more than 37 minutes

Drive time more than 44 minutes

Drive time more than 49 minutes

We know that, a z-score is unlikely, if it is less than -2 or greater than 2.

So, in this problem, the unlikely z-scores are

z<-2.8

z>2.4

z>3.4

The corresponding drive times are unlikely.

So, the answer is

The highly unlikely droive times are

option (A) less than 18 minutes.

option (E) more than 44 minutes.

option (F) more than 49 minutes.

Question 4)

We have to find, longest 4% of the drives require at least how many minutes.

Now, if m is the answer, then it is given that

Now, from the standard normal table, we note that

So, we can conclude that

Rounding off to one decimal place, m becomes 40.8.

The answer is

The 4% longest drives require at least 40.8 minutes.