Question

Floretta's points per basketball game are normally distributed with a standard deviation of 4 points. If Floretta scores 10

points, and the Z-score of this value is - 4, then what is her mean points in a game? Do not include the units in your answer.
For example, if you found that the mean is 33 points, you would enter 33.

Answer 1

26

Explanation:

We can work backwards using the z-score formula to find the mean. The problem gives us the values for z, x and σ. So, let's substitute these numbers back into the formula:

z−4−16−2626=x−μσ=10−μ4=10−μ=−μ=μ

We can think of this conceptually as well. We know that the z-score is −4, which tells us that x=10 is four standard deviations to the left of the mean, and each standard deviation is 4. So four standard deviations is (−4)(4)=−16 points. So, now we know that 10 is 16 units to the left of the mean. (In other words, the mean is 16 units to the right of x=10.) So the mean is 10+16=26.

Answer 2

`26` points.

Explanation:

Let
`X` denote a normal random variable with mean
`\mu` and standard deviation
`\sigma`. (That is:
`X \sim {\rm N}(\mu,\, \sigma)`.) By definition, the
`z`-score of an observation with value
`X = x` would be:

`\begin{aligned} z&= (x - \mu)/(\sigma)\end{aligned}`.

In this question, the value of
`\sigma` is given. Also given are the value of the observation
`x` and the corresponding
`z`-score,
`z\!`. Rearrange the
`\! z`-score definition
`z = (x - \mu) / \sigma` to find an expression for
`\mu`:

`\begin{aligned} x - \mu = \sigma\, z\end{aligned}`.

`-\mu = (-x) + \sigma\, z`.

`\begin{aligned}\mu = x - \sigma\, z\end{aligned}`.

Substitute in the value of
`x`,
`\sigma`, and
`z` to find the value of
`\mu`, the mean of this normal random variable:

`\begin{aligned}\mu &= x - \sigma\, z \\ &= 10 - (-16) \\ &= 26\end{aligned}`.