Question

(a) judy has not taken a statistics class in a few years. explain to her in simple language what the standardized score reveals about her bone density. (b) use the information provided to calculate the standard deviation of bone density in the reference population.

Explanation:

Step1: Define standardized score

The standardized score (z - score) is calculated as $z=\frac{x - \mu}{\sigma}$, where $x$ is a data - point, $\mu$ is the mean of the population, and $\sigma$ is the standard deviation of the population. It tells us how many standard deviations a particular value ($x$) is away from the mean ($\mu$). For Judy's bone density, if her standardized score is positive, her bone density is above the mean bone density of the reference population. If it is negative, her bone density is below the mean. If it is zero, her bone density is equal to the mean of the reference population.

Step2: Rearrange z - score formula for standard deviation

We know the z - score formula $z=\frac{x - \mu}{\sigma}$, and we want to find $\sigma$. Rearranging the formula for $\sigma$ gives $\sigma=\frac{x - \mu}{z}$. We need to know the value of Judy's bone density ($x$), the mean bone density of the reference population ($\mu$), and her standardized score ($z$) to calculate the standard deviation of the bone density in the reference population.

However, since no values for $x$, $\mu$ and $z$ are provided in the question for part (b), we can only give the formula for calculating the standard deviation.

Answer:

(a) The standardized score tells Judy how many standard deviations her bone density is from the mean bone density of the reference population. A positive z - score means her bone density is above the mean, a negative z - score means it is below the mean, and a z - score of 0 means it is equal to the mean.
(b) $\sigma=\frac{x - \mu}{z}$ (where $x$ is Judy's bone density, $\mu$ is the mean bone density of the reference population, and $z$ is her standardized score)