Answer:
To answer these questions, we'll use the Z-score formula and the properties of the standard normal distribution (z-distribution), which has a mean (μ) of 0 and a standard deviation (σ) of 1.
a. What mileage corresponds to a Z-score of 2.5?
To find the mileage corresponding to a Z-score of 2.5, we'll use the formula for the Z-score:
Z = (X - m) / σ
Where:
Z is the Z-score (given as 2.5).
X is the value we want to find (the mileage).
μ is the mean (9750 miles).
σ is the standard deviation (3820 miles).
Let's rearrange the formula to solve for X:
X = m + Z * σ
X = 9750 + 2.5 * 3820
X = 9750 + 9550
X = 19,300 miles
So, a Z-score of 2.5 corresponds to a mileage of approximately 19,300 miles per year.
b. What mileage corresponds to a Z-score of 1.25?
Using the same formula:
X = m + Z * σ
X = 9750 + 1.25 * 3820
X = 9750 + 4775
X = 14,525 miles
A Z-score of 1.25 corresponds to a mileage of approximately 14,525 miles per year.
c. Approx 95% of US adults aged 21+ drive between and miles per year.
To find the range of miles driven that corresponds to approximately 95% of the distribution, we can use the properties of the standard normal distribution. In the standard normal distribution:
About 68% of the data falls within one standard deviation (σ) of the mean (μ).
About 95% of the data falls within two standard deviations (σ) of the mean (μ).
So, for our problem:
The mean (μ) is 9750 miles.
The standard deviation (σ) is 3820 miles.
To find the range, we'll go two standard deviations above and below the mean:
Lower limit: 9750 - 2 * 3820 = 2110 miles
Upper limit: 9750 + 2 * 3820 = 17,390 miles
So, approximately 95% of US adults aged 21+ drive between 2,110 and 17,390 miles per year.
Explanation: