In an analysis of healthcare data, ages have been rounded to the nearest multiple of 5 years. The difference between the true age and the rounded age is assumed to be uniformly distributed on the interval from - 2.5 years to 2.5 years. The healthcare data are based on a random sample of 48 people.

Calculate the approximate probability that the mean of the rounded ages is within 0.25 years of the mean of the true ages.

• 0.14
• 0.38
• 0.57
• 0.77
• 0.88

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

For one observation, the mean is 0 and the variance is 25/12 (for a uniform distribution the variance is the square of the range divided by 12). For 48 observations, the average has a mean of 0 and a variance of (25/12)/48 = 0.0434. The standard deviation is 0.2083. 0.25 years is 0.25/0.2083 = 1.2 standard deviations from the mean. From the normal tables the probability of being within 1.2 standard deviations is 0.8849 – (1 – 0.8849) = 0.7698.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.