The average age of a vehicle registered in the United States is 8 years, or 96 months. Assume the standard deviation is 16 months. If a random sample of a group of 48 vehicles is selected find the probability that the mean of the group’s age is between 90 and 100 months.
Z = (score-mean)/SEm
SEm = SD/√n
Use same table indicated in previous post.
To find the probability that the mean of the group's age is between 90 and 100 months, we need to use the normal distribution and calculate the z-scores.
1. Find the z-score for the lower limit (90 months):
z1 = (90 - 96) / (16 / sqrt(48))
= -6 / (16 / sqrt(48))
≈ -1.94
2. Find the z-score for the upper limit (100 months):
z2 = (100 - 96) / (16 / sqrt(48))
= 4 / (16 / sqrt(48))
≈ 1.23
3. Look up the corresponding probabilities for the z-scores using a standard normal distribution table or a calculator.
P(z < -1.94) ≈ 0.0274
P(z < 1.23) ≈ 0.8907
4. Subtract the probability of the lower limit from the probability of the upper limit to get the probability that the mean is between 90 and 100 months:
P(-1.94 < z < 1.23) ≈ P(z < 1.23) - P(z < -1.94)
≈ 0.8907 - 0.0274
≈ 0.8633
Therefore, the probability that the mean of the group's age is between 90 and 100 months is approximately 0.8633 or 86.33%.