1. The problem is to convert random variables to standard normal values (z-scores) and convert standard normal values back to original random variables. 2. The formula to convert a random variable $X$ with mean $\mu$ and standard deviation $\sigma$ to a standard normal variable $Z$ is: $$Z = \frac{X - \mu}{\sigma}$$ This formula standardizes $X$ by subtracting the mean and dividing by the standard deviation. 3. To convert a standard normal variable $Z$ back to the original variable $X$, use: $$X = Z \times \sigma + \mu$$ This reverses the standardization by multiplying by the standard deviation and adding the mean. 4. Important rules: - The mean $\mu$ is the average value of the original variable. - The standard deviation $\sigma$ measures the spread of the original variable. - The standard normal variable $Z$ has mean 0 and standard deviation 1. 5. Example: Suppose $X$ has mean $\mu=50$ and standard deviation $\sigma=10$. - To convert $X=70$ to $Z$: $$Z = \frac{70 - 50}{10} = \frac{20}{10} = 2$$ - To convert $Z=2$ back to $X$: $$X = 2 \times 10 + 50 = 20 + 50 = 70$$ This shows how to switch between original and standard normal values easily. Final answer: Use $$Z = \frac{X - \mu}{\sigma}$$ to convert to standard normal and $$X = Z \times \sigma + \mu$$ to convert back.