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? (1 pt) b) less than 11.4 minutes? (1 pt) c) greater than 37.4 minutes? (1 pt) 3. Using the parameters provided in the previous question, Q2, determine the z-score that corresponds to a commute time of 15 minutes. (2 pts)

Answer 1

Answer:

Q1 a) 0.9545

b) 0.02275

c) 0.97725

Q2 z is approximately equal to -1.466

Step-by-step explanation:

Q1 The given information are;

The mean time to commute to work = 24.4 minutes

The standard deviation = 6.5 minutes

a) The z-score for 11.4 is given as follows;

$Z=\dfrac{x-\mu }{\sigma }$

Where;

x = Observed value 11.4

μ = The mean = 24.4 minutes

σ = The standard deviation = 6.5 minutes

$Z=\dfrac{11.4-24.4 }{6.5 } = -2$

The z-score for 37.4 is given as follows;

$Z=\dfrac{37.4-24.4 }{6.5 } = 2$

-2 < z < 2, which gives, from the z-score table;

The probability of commute time to be between 11.4 minutes and 37.4 minutes =  0.97725 - 0.02275 = 0.9545

b) From the z-score table, the probability that the commute time to be less than 11.4 minutes = The probability at z = -2 = 0.02275

c) From the z-score table, the probability that the commute time to be greater than 37.4 minutes = The probability at z = 2 = 0.97725

Q2 The the z-score that corresponds to a commute time of 15 minutes is given as follows;

$Z=\dfrac{x-\mu }{\sigma }$

$Z=\dfrac{15-24.4 }{6.5 } = -\dfrac{94}{65} \approx -1.466$