The average commute time via train from the Chicago O'Hare Airport to downtown is 60 minutes with a s=15 minutes. Assume that the commute times are normally distributed. what proportion of commutes would be:
Between 45 and 75 minutes?
To find the proportion of commutes that would be between 45 and 75 minutes, we need to calculate the area under the normal distribution curve between these two points.
First, we need to calculate the z-scores for the lower and upper bounds of the range.
The z-score formula is given by (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
For the lower bound of 45 minutes:
z = (45 - 60) / 15 = -1
For the upper bound of 75 minutes:
z = (75 - 60) / 15 = 1
Next, we need to use a standard normal distribution table or a calculator to find the corresponding probabilities for these z-scores.
Looking up the z-score of -1 in the table, we find that the corresponding probability is 0.1587.
Looking up the z-score of 1 in the table, we find that the corresponding probability is 0.8413.
To find the proportion of commutes that would be between 45 and 75 minutes, we subtract the lower probability from the upper probability:
0.8413 - 0.1587 = 0.6826
Therefore, approximately 68.26% of commutes would be between 45 and 75 minutes.
To find the proportion of commutes between 45 and 75 minutes, we need to calculate the z-scores for both values and then use the z-score table to find the corresponding proportions.
First, let's calculate the z-scores:
For 45 minutes:
z = (45 - 60) / 15 = -1
For 75 minutes:
z = (75 - 60) / 15 = 1
Next, we look up the proportions corresponding to these z-scores in the z-score table.
The z-score table shows that for a z-score of -1, the corresponding proportion is 0.1587 (or 15.87%).
For a z-score of 1, the corresponding proportion is 0.8413 (or 84.13%).
To find the proportion of commutes between 45 and 75 minutes, we subtract the proportion for the lower bound from the proportion for the upper bound:
Proportion = 0.8413 - 0.1587 = 0.6826
Therefore, approximately 68.26% of commutes would be between 45 and 75 minutes.