bell shaped distribution

Question

The commute time to work in the U.S. has a bell shaped distribution with a population mean of 24.4 minutes and a population standard deviation of 6.5 minutes. What proportion of the population has a commute time:

a) between 11.4 minutes and 37.4 minutes?

b) less than 11.4 minutes? 

c) greater than 37.4 minutes? 

3. Using the parameters provided in the previous question, Q2, determine the z-score that corresponds to a commute time of 15 minutes.

Answer 1

Let x be the commute time. x is normally distributed with mean 24.4 and standard deviation 6.5

For part a) we want to find P(11.4 < x < 37.4)

If we convert the "x" range to "z-scores" we can use the standard normal probabilities table.

A z-score is equal to (x - mean) / standard deviation = (x - 24.4)/6.5

If we convert the above probability statement to z-scores we get:

P( (11.4-24.4)/6.5 < z < (37.4-24.4)/6.5)

= P(-2 < z < 2)

= P(z < 2) - P(z < -2)

The above probabilities can be found from the standard normal probabilities table:

= 0.9772 - -0.0228 = 0.9544

For part b) we want to find P(x < 11.4)

From part a) we can convert that to P(z < -2) = 0.0228

For part c) we want to find P(x > 37.4)

From part a) we can convert that to P(z > 2). Since the normal distribution is symmetric that is the same as P(z < -2) = 0.0228. Or you could compute it as 1 - P(z < 2) = 1 - 0.9772 = 0.0228.