Question

The 16th percentile of a Normally distributed variable has a value of 25 and the 97.5 th percentile has a value of 40 . Which of the following is the best estimate of the mean and standard deviation of the variable?

A. Mean ≈32.5; Standard deviation ≈2.5
B. Mean ≈32.5; Standard deviation ≈5
C. Mean ≈32.5; Standard deviation ≈10
D. Mean ≈30; Standard deviation ≈2.5
E. Mean ≈30; Standard deviation ≈5

Answer 1

To estimate the mean and standard deviation of a normally distributed variable, use the z-score formula with the given percentiles. The best estimate of the mean is approximately 32.5, and the best estimate of the standard deviation is approximately 5.

Step-by-step explanation:

To estimate the mean and standard deviation of the normally distributed variable, we can use the z-score formula. The z-score is the number of standard deviations away from the mean a data point is. We can use the given percentiles to calculate the z-scores:

For the 16th percentile: z = (X - mean) / standard deviation, we have 25 = (X - mean) / standard deviation.

For the 97.5th percentile: z = (X - mean) / standard deviation, we have 40 = (X - mean) / standard deviation.

Solving these two equations, we can find the values of the mean and standard deviation.

The best estimate of the mean is approximately 32.5, and the best estimate of the standard deviation is approximately 5.