$7\mathrm{.}13$ The Environmental Protection Agency is concerned with the problem of serting criteria for the amountx of certain toxic chemicals to be allowed in freshwater lakes and rivers. A common measure of toxicity for any pollutant is the concentration of the poliutant that will kill half of the test species in a given amount of time $($usually $96$ hours for fish species$).$ This measure is called LC$50($lethal concentration killing S$0\%$ of the test species$),$ in many studies, the values contained in the natural logarithm of LCSO measurements are normally distributed, and, hence, the analysis is based on In$($LC$50)$ data.

Studies of the effects of copper on a certiain species of fish $($siay species A$)$ show the variance of $ln(LCSO)$ measurements to be around $.4$ with concentration measurements in milligrams per liter. If $n=10$ studies on LC$50$ for copper are to be completed, find the probability that the sample mean of $ln(LCSO)$ will differ from the true population mean by no more than $5.$