Question

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

Answer 1

Answer:

$26$ points.

Step-by-step 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:

$\begin{aligned} z&= \frac{x - \mu}{\sigma}\end{aligned}$.

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:

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

Answer 2

Answer:

26

Step-by-step 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.