Question: Dioxin study. Dioxin, often described as the most toxic chemical known, is created as a by-product in the manufacture of herbicides such as Agent Orange. Scientists have found that .000005 gram (five-millionths of a gram) of dioxin – a dot barely visible to the human eye – is a lethal dose for experimental guinea pigs in more than half the animals tested, making dioxin 2,000 times more toxic than strychnine. Assume that the amount of dioxin required to kill a guinea pig was a relative frequency distribution with mean \[\mu =.000005\text{ gram and standard deviation }\sigma \text{=}\text{.000002 gram}\text{.}\] Consider an experiment in which the amount of dioxin required to kill each of \[n=50\] guinea pigs is measured, and the sample mean \[\overline{Y}\] is computed.

a. Calculate \[{{\mu }_{\overline{Y}}}\text{ and }{{\sigma }_{\overline{Y}}}\].

b. Find the probability that the mean amount of dioxin required to kill the 50 guinea pigs is larger than .0000053 gram.