1. ## Sampling Distribution Question

I'm trying to help my girlfriend with this problem however its been quite a while since I took stats and I'm hoping you can help it reads:

Last year a company initiated a program to compensate its employees for unused sick days, paying each employee a bonus of one-half the usual wage earned for each unused sick day. The questions the naturally arises is "Di this policy motivate employees to use fewer allotted sick days?"

Before last year, employees averaged 7 sick days per year with a standard deviation of 2. Assuming these parameters did not change last year, find the approximate probability that the sample mean number of sick days used by 100 employees chosen at random was less than or equal to 6.4 last year.

And, suppose the sample mean for the 100 employees was in fact 6.4 how would you interpret this result?

Thank you

Chris

2. using the central limit theorum sample means are approximately normal with mean $\mu$ and variance $\frac{\sigma^2}{n}$

so
$X \sim N(7,4/100)$

find $P(X \leq 6.4) = P(Z \leq \frac{6.4 - 7}{2/10}) = \Phi(-3) = ...$

This tells you how likely you are to obvserve a mean of less than 6.4 if the bonus had no effect. Use this to reach a conclusion in the last part.

3. thank you