Kumail K. answered • 03/18/23
Online Exam and Quiz Expert with Proven Results
Hi student check my answer:
Part (a):
The minimum percentage of commuters in Louisville who have a commute time within 2 standard deviations of the mean is at least 95%. This is because the range within 2 standard deviations of the mean includes 95% of the total area under the normal curve.
To calculate this, we first find the range within 2 standard deviations of the mean using the formula:
Mean ± 2(Standard Deviation) = 23.9 ± 2(5.2) = 13.5 to 34.3. This means that the commute time for at least 95% of the commuters falls within the range of 13.5 to 34.3 minutes. Part (b):
To find the percentage of commuters in Louisville who have a commute time of less than 30 minutes, we need to calculate the z-score for 30 using the formula:
z = (x - μ) / σ
where x is the value we want to find the z-score for, μ is the mean, and σ is the standard deviation.
So, for x = 30, we have:
z = (30 - 23.9) / 5.2 = 1.17. Using a standard normal distribution table or a calculator, we can find the area under the normal curve to the left of z = 1.17, which is approximately 0.8790 or 87.90%.
Therefore, approximately 87.90% of the commuters in Louisville have a commute time of less than 30 minutes.
Part (c):
To find the commute time for the top 5% of the commuters in Louisville, we need to find the z-score such that the area under the normal curve to the right of that z-score is 0.05. Using a standard normal distribution table or a calculator, we can find that the z-score is approximately 1.645.
So, we can use the z-score formula to find the commute time corresponding to this z-score:
z = (x - μ) / σ
1.645 = (x - 23.9) / 5.2
x - 23.9 = 8.554
x = 32.454
Therefore, the commute time for the top 5% of the commuters in Louisville is approximately 32.454 minutes.
