How do I find the $Z$ value and calculate the $P$?

Question

Please DO NOT answer the question, as I just need the right formulas in the form of plugging the numbers from the word problem into it. I struggle with this. And I need to find what the percentage is of the money she makes compared to other babysitters. Please help.

Your babysitter claims that she is underpaid given the current market. Her hourly wage is $12$ per hour. The normal average is $14$ per hour with a standard deviation of $1.9$. Calculate the $Z$ score and use the table to find the standard normal probability. Based on your findings, should you give her a raise? Explain your reasoning as to why or why not.

Answer 1

If you have a particular value $x$, and you want to standardize it, you calculate its $z$ score $$z_{\text{score}} = \frac{x-\mu}{\sigma}$$ where $\mu$ is the population average and $\sigma$ is the population standard deviation.

Depending on the table you use, the $z$ score will help you find the area under the curve of a standard normal distribution.

For example, if your table is defined as "area to the left", then the value of the table $\Phi(z_\text{sc})$ will give the area under the standard normal curve from $-\infty$ to $z_{\text{sc}}$. This area is a value in $(0,1)$ and can be interpreted as a percentage.

Answer 2

In layman's terms (somewhat), the standard score formula (or $z$-score) is $$z := \frac{x-\mu}{\sigma},$$ where $\mu$ is the (population/sample) mean, $\sigma$ is the (population/sample) standard deviation, and $x$ (called the score) is the value you want to calculate the standard score for. Once you obtain the value, $z$, you can look it up in this table to find what you call $P$ in your question (it's actually the area under the standard normal distribution curve from $-\infty$ to $z$ according to this table, that is, $P(Z\leq z)$ where $Z\sim \text{Normal}(0,1)$).