Question

In a population of exam scores, a score of x=48 corresponds to z=+1.00 and a score of x=36 corresponds to z=-0.05. Find the mean and standard deviation for the population.
hint. Sketch the distribution and locate the two scores on your sketch.

Answer 1

Answer:

$\mu=36.57$

$\sigma=11.43$

Step-by-step explanation:

We have been given that in a population of exam scores, a score of x=48 corresponds to z=+1.00 and a score of x=36 corresponds to z=-0.05.

To find the values of mean and standard deviation, we will use z-score formula as:

$z=\frac{x-\mu}{\sigma}$

Let us solve for x (sample score).

$x=z\cdot \sigma+\mu$

Upon substituting our given values, we will get two equations as:

$48=1\cdot \sigma+\mu...(1)$

$48-\mu=\sigma...(1)$

$36=-0.05\cdot \sigma+\mu...(2)$

Upon substituting equation (1) in equation (2), we will get:

$36=-0.05(48-\mu)+\mu$

$36=-2.4+0.05\mu+\mu$

$36+2.4=1.05\mu$

$38.4=1.05\mu$

$\frac{38.4}{1.05}=\frac{1.05\mu}{1.05}$

$\mu=36.57142\approx 36.57$

Therefore, the mean of the given data would be approximately 36.57.

Upon substituting $\mu=36.57$ in equation (1), we will get:

$48-36.57=\sigma$

$\sigma=11.43$

Therefore, the standard deviation for the given data would be approximately 11.43.