Question

1. A normal distribution of scores has a standard deviation of 10. Find the z-scores corresponding to each of the following values: a) A score that is 20 points above the mean. b) A score that is 10 points below the mean. c) A score that is 15 points above the mean d) A score that is 30 points below the mean. 2. The Welcher Adult Intelligence Test Scale is composed of a number of subtests. On one subtest, the raw scores have a mean of 35 and a standard deviation of 6. Assuming these raw scores form a normal distribution:

Answer 1

Answer with Step-by-step explanation:

The Z score in a distribution is found as

$Z=\frac{X-\overline{X}}{\sigma }$

where

X is the given value

$\overline{X}$ is the mean of the observations

$\sigma$ is the standard deviation of the data

The values of oach part is found as under:

Part a)

Since it is given that score is 20 points over mean thus we can write

$X=20+\overline{X}\\\\X-\overline{X}=20$

Using the above relation we get

$Z_a=\frac{20}{10}=2$

Part b)

Since it is given that score is 10 points below mean thus we can write

$X=\overline{X}-10\\\\X-\overline{X}=-10$

Using the above relation we get

$Z_b=\frac{-10}{10}=-1$

Part c)

Since it is given that score is 15 points over mean thus we can write

$X=15+\overline{X}\\\\X-\overline{X}=15$

Using the above relation we get

$Z_c=\frac{15}{10}=1.5$

Part d)

Since it is given that score is 30 points below mean thus we can write

$X=\overline{X}-30\\\\X-\overline{X}=-30$

Using the above relation we get

$Z_d=\frac{-30}{10}=-3$