This is one of our homework problems for Statistics. I'm really stuck on this one, and it would be great if someone could show me how to solve this, or at least set up the equation.
The weights of chickens on old McDonald’s farm are normally distributed with a standard deviation of 4.2 lb. Find the mean weight of old McDonald’s chickens if only 4% weigh less than 20 lbs.
Best Answer
Let the weight of the chickens be $X$. Then $X \sim N(\mu,\sigma^2)$ where $\sigma = 4.2$ is the standard deviation of the weights and $\mu$ is the mean of the weights. Then, the statement you're given is $P(X < 20) = 0.04$.
Now, rewrite $P(X<20) = P(X-\mu < 20-\mu) = P( \frac{X-\mu}{\sigma} < \frac{20-\mu}{\sigma}) = P(Z < \frac{20-\mu}{\sigma})$ where $Z \sim N(0,1)$. Then, look up $c$ in a standard normal table such that $P(Z<c) = 0.04$ and then note $\frac{20-\mu}{\sigma} = c$. Solve for $\mu$, and you are done.