The average age of a vehicle registered in the United States is 8 years, or 96 months. If a random sample of 36 vehicles is selected, find the probability that the mean of their age is between 98 and 100 months. Assume the standard deviation for the population is 15.
First, we need to find the standard deviation of the sample mean. The standard deviation of the sample mean, also known as the standard error, can be calculated using the formula:
standard error = standard deviation / sqrt(sample size)
In this case, the standard deviation is 15 and the sample size is 36:
standard error = 15 / sqrt(36)
standard error = 15 / 6
standard error = 2.5
Now, we need to find the z-scores for the lower and upper limits of the desired range. We can use the formula:
z = (x - μ) / standard error
For the lower limit:
z1 = (98 - 96) / 2.5
z1 = 0.8
For the upper limit:
z2 = (100 - 96) / 2.5
z2 = 1.6
Next, we need to find the probabilities associated with these z-scores using a standard normal distribution table or calculator. The probability that the mean of the sample falls between 98 and 100 months can be calculated as:
P(98 < x < 100) = P(z1 < z < z2)
Using a standard normal distribution table or calculator, we can find the probabilities associated with z1 and z2:
P(z < 0.8) = 0.7881 (approximately)
P(z < 1.6) = 0.9452 (approximately)
Now, we can calculate the desired probability:
P(98 < x < 100) = P(z1 < z < z2)
P(98 < x < 100) = P(z < 1.6) - P(z < 0.8)
P(98 < x < 100) = 0.9452 - 0.7881
P(98 < x < 100) = 0.1571
Therefore, the probability that the mean of the vehicle ages is between 98 and 100 months is approximately 0.1571, or 15.71%.