Question

given that z_20 = -2 and z_50 = -1, which of the following do you know? the variance is 10. the standard deviation is 30. the mean is 80. the median is 40. the data point x = 20 is 2 standard deviations from the mean. the data point x = 50 is 1 standard deviation from the mean. the data point x = 45 has a z - value of 1.5.

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x - \mu}{\sigma}$, where $z$ is the z - score, $x$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation. Given $z_{20}=-2$, we have $-2=\frac{20 - \mu}{\sigma}$, which implies $20-\mu=-2\sigma$. Given $z_{50}=-1$, we have $-1=\frac{50 - \mu}{\sigma}$, which implies $50-\mu=-\sigma$.

Step2: Analyze z - score meaning

The z - score represents the number of standard deviations a data point is from the mean. A z - score of $z_{20}=-2$ means the data point $x = 20$ is 2 standard deviations below the mean. A z - score of $z_{50}=-1$ means the data point $x = 50$ is 1 standard deviation below the mean.
We do not have enough information to find the variance, standard deviation, mean, or median from just the two z - scores.

Answer:

The data point $x = 20$ is 2 standard deviations from the mean.
The data point $x = 50$ is 1 standard deviation from the mean.