The Superintendent of a Large School District, Having Once Had $\lambda$

question 30

Essay

The superintendent of a large school district, having once had a course in probability and statistics, believes that the number of teachers absent on any given day has a Poisson distribution with parameter $\lambda$ . Use the accompanying data on absences for 50 days to derive a large-sample CI for $\lambda$ . [Hint: The mean and variance of a Poisson variable both equal $\lambda$ , so $Z = ( \bar{X} - \lambda ) / \sqrt { \lambda / n }$ has approximately a standard normal distribution. Now proceed as in the derivation of the interval for p by making a probability statement (with probability 1 - $\alpha$ ) and solving the resulting inequalities for $\lambda$ . $$\begin{array} { c c c c c c c c c c c c } \hline \text { \#Absences } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline \text { Frequency } & 1 & 6 & 8 & 10 & 8 & 7 & 5 & 3 & 2 & 1 & 1 \\\hline\end{array}$$

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Definitions:

Accumulated Benefit Obligation

The actuarial present value of all future pension benefits earned to date, based on current salary levels, without assuming future salary increases.

Pension Liability

The financial obligation a company has to provide its employees with pension benefits, representing the difference between the total amount due to retirees and the amount of funds the company has set aside to cover these costs.

Year-End Balance Sheet

A financial statement that provides a snapshot of a company's financial condition at the end of a fiscal year, detailing assets, liabilities, and shareholders' equity.

Pension Liability

The total amount of money that a company is obligated to pay into its current and future retirees' pension plans.