Question

the mean score on a physics test was 75 points. amy’s score was 67 points, which was 2 standard deviations below the mean. what is the variance of the data set? 2 4 8 16

Explanation:

Step1: Recall the z - score formula

The z - score formula is \(z=\frac{x - \mu}{\sigma}\), where \(z\) is the number of standard deviations from the mean, \(x\) 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 is 2 standard deviations below the mean, the z - score is negative).

Step2: Solve for the standard deviation \(\sigma\)

Substitute the known values into the z - score formula: \(-2=\frac{67 - 75}{\sigma}\)
First, simplify the numerator: \(67-75=-8\)
So the equation becomes \(-2=\frac{-8}{\sigma}\)
Cross - multiply: \(-2\times\sigma=-8\)
Divide both sides by \(- 2\): \(\sigma=\frac{-8}{-2} = 4\)

Step3: Calculate the variance

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

Answer:

16