1. 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?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to your Z scores.
To answer these questions, we will need to use the concept of the standard normal distribution.
The standard normal distribution, also known as the Z-distribution, is a bell-shaped curve with a mean of 0 and a standard deviation of 1. By standardizing the values in our problem, we can use a standard normal distribution table (also known as a Z-table) to find the proportion of values that fall within a specific range.
To convert our commute time values to Z-scores, we use the formula:
Z = (X - μ) / σ
where Z is the Z-score, X is the value we want to convert, μ is the mean, and σ is the standard deviation.
Now let's calculate the proportions for each of the given situations:
a. To find the proportion of commutes longer than 80 minutes, we first need to convert 80 minutes to a Z-score:
Z = (80 - 60) / 15
Z = 1.33
Next, we look up the corresponding area under the standard normal curve for a Z-score of 1.33 in the tail of the distribution. Using a standard normal distribution table or calculator, we find that the proportion is approximately 0.0918 or 9.18%.
b. To find the proportion of commutes less than 50 minutes, we convert 50 minutes to a Z-score:
Z = (50 - 60) / 15
Z = -0.67
Again, we look up the area under the standard normal curve for a Z-score of -0.67 in the body of the distribution (the area to the left of the Z-score). This proportion is approximately 0.2514 or 25.14%.
c. To find the proportion of commutes between 45 and 75 minutes, we need to find the individual proportions for each value and subtract them:
Proportion less than 75 minutes:
Z = (75 - 60) / 15
Z = 1
Proportion less than 45 minutes:
Z = (45 - 60) / 15
Z = -1
Now we look up the areas under the standard normal curve for Z = 1 and Z = -1. The proportion for Z = 1 is approximately 0.8413 and the proportion for Z = -1 is approximately 0.1587. Subtracting the two proportions gives:
0.8413 - 0.1587 = 0.6826 or 68.26%
Therefore, approximately 68.26% of commutes would fall between 45 and 75 minutes.
These calculations assume that the commute times are normally distributed.