2. The average commute time via train from the Chicago O'Hare Airport to downtown is 60 minutes with a standard deviation of 15 minutes. Assume that the commute times are normally distributed. What proportion of commutes would be:
a. longer than 80 minutes?
b. less than 50 minutes?
c. between 45 and 75 minutes?
To find the proportions of commutes for different time intervals, we need to use the normal distribution and apply the Z-score formula. The Z-score measures the number of standard deviations an observation is from the mean.
First, we need to calculate the Z-scores for the given time intervals using the formula:
Z = (X - μ) / σ
Where:
X = the value we want to find the proportion for (in minutes)
μ = the mean commute time (in minutes)
σ = the standard deviation of commute times (in minutes)
a. To find the proportion of commutes longer than 80 minutes:
Z = (80 - 60) / 15 = 20 / 15 = 1.33
Using a Z-table or a calculator, we can find the proportion of values greater than 1.33, which indicates the area under the curve to the right of 1.33. This proportion represents the commutes that are longer than 80 minutes.
b. To find the proportion of commutes less than 50 minutes:
Z = (50 - 60) / 15 = -10 / 15 = -0.67
Using a Z-table or a calculator, we can find the proportion of values less than -0.67, which indicates the area under the curve to the left of -0.67. This proportion represents the commutes that are less than 50 minutes.
c. To find the proportion of commutes between 45 and 75 minutes:
To find the proportion between two values, we need to subtract the area under the curve to the left of the lower limit from the area under the curve to the left of the upper limit.
Z for the lower limit (45 minutes):
Z_lower = (45 - 60) / 15 = -15 / 15 = -1
Z for the upper limit (75 minutes):
Z_upper = (75 - 60) / 15 = 15 / 15 = 1
Now, using a Z-table or a calculator, we can find the area to the left of -1 and the area to the left of 1, and subtract the smaller area from the larger area to find the proportion of commutes between 45 and 75 minutes.
Note: If you are using a Z-table, you will have to convert the Z-scores to proportions by looking up the values in the table. Alternatively, you can use a calculator or an online tool to find the proportions directly.