Question

the mean score on a physics test was 75 points. amys score was 57 points, which was 2 standard deviations below the mean. what is the variance of the data - set? o 2 o 4 o 8 o 16

Explanation:

Step1: Set up the z - score formula

The z - score formula is $z=\frac{x - \mu}{\sigma}$, where $x$ is the data - point, $\mu$ is the mean, and $\sigma$ is the standard deviation. We know that $x = 67$, $\mu=75$, and $z=- 2$ (since it's 2 standard deviations below the mean). Substituting these values into the formula gives $-2=\frac{67 - 75}{\sigma}$.

Step2: Solve for the standard deviation $\sigma$

Cross - multiply the equation $-2=\frac{67 - 75}{\sigma}$: $-2\sigma=67 - 75$. Then $-2\sigma=-8$, so $\sigma = 4$.

Step3: Calculate the variance

The variance $\sigma^{2}$ is the square of the standard deviation. Since $\sigma = 4$, then $\sigma^{2}=16$.

Answer:

D. 16