Sample Mean

Study guide: Exam MAS-I

The Sample Mean \bar{X} is the arithmetic average of a random sample and the standard estimator of the population mean \mu. It is unbiased, consistent, and — for a Normal population — the Minimum Variance unbiased estimator.

\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i

E[\bar{X}] = \mu, \qquad \text{Var}(\bar{X}) = \frac{\sigma^2}{n}

• The standard error is \text{SD}(\bar{X}) = \sigma/\sqrt{n}; estimated from data it is S/\sqrt{n}. Halving it requires four times the sample
• By the Central Limit Theorem, \bar{X} is approximately N(\mu, \sigma^2/n) for large n whatever the population shape — the fact behind almost every test and Confidence Interval in this section
• For a Normal population the result is exact: \bar{X} \sim N(\mu, \sigma^2/n), and \bar{X} is independent of the Sample Variance S^2
• \bar{X} is the MLE of \mu for Normal, Poisson, and exponential data, and is a function of the Sufficient Statistic \sum X_i in each case
• It is not resistant: one extreme claim shifts \bar{X} without limit, while the median and the other quantiles a Box Plot is drawn from are relatively insensitive to changes in the data
• With unequal exposures, the exposure-weighted mean \sum w_i X_i / \sum w_i is the right estimator — the same idea that a GLM handles with an offset or weights

Standard Error and Sample Size Claim severity has \sigma = 4{,}000. How large must n be for the standard error of \bar{X} to be at most 200?

Answer \frac{\sigma}{\sqrt{n}} \le 200 \ \Longrightarrow\ \sqrt{n} \ge \frac{4{,}000}{200} = 20 \ \Longrightarrow\ n \ge 400 A sample of 400 claims pins the mean down to within roughly \pm 400 (two standard errors) with 95\% confidence.

Probability the Sample Mean Misses A sample of n = 100 claims is drawn from a population with \mu = 5{,}000 and \sigma = 1{,}500. Find P(\bar{X} > 5{,}300).

Answer By the Central Limit Theorem, \bar{X} \approx N(5{,}000,\ 1{,}500^2/100) = N(5{,}000,\ 150^2). Z = \frac{5{,}300 - 5{,}000}{150} = 2.00 \ \Longrightarrow\ P(\bar{X} > 5{,}300) \approx 1 - 0.9772 = 0.0228