Question

Current research indicates that the distribution of the life expectancies of a certain protozoan is normal with a mean of 48 days and a standard deviation of 10.2 days. find the probability that a simple random sample of 49 protozoa will have a mean life expectancy of 49 or more days.

Answer 1

`X` is the random variable for lifespan of a protozoan and
`X\sim\mathcal N(48,10.2^2)`. Let
`\bar X` be the mean of a sample from this distribution, so that
`\bar X\sim\mathcal N\left(48,\left((10.2)/(√(49))\right)^2\right)`.

For the sake of clarity, I'm denoting a normal distribution with mean
`\mu` and standard deviation
`\sigma` by
`\mathcal N(\mu,\sigma^2)`.

We have

(where
`Z` follows the standard normal distribution)