On a standard measure of hearing ability, the mean is 300 and the standard deviation is 20. Give the Z scores for persons who score (a) 340, (b) 310, and (c) 260. Give the raw scores for persons whose Z scores on this test are (d) 2.4, (e) 1.5, (f) 0, and (g) –4.5.
z = (value - mean)/SD
I'll do one of each type. It requires only algebra once you have the equation.
a. z = (340 - 300)/20
z = 2
d. 2.4 = (value - 300)/20
48 = value - 300
value = 348
To find the Z score, we need to use the formula:
Z = (X - μ) / σ
where X is the raw score, μ is the mean, and σ is the standard deviation.
(a) For a score of 340:
Z = (340 - 300) / 20
Z = 40 / 20
Z = 2
(b) For a score of 310:
Z = (310 - 300) / 20
Z = 10 / 20
Z = 0.5
(c) For a score of 260:
Z = (260 - 300) / 20
Z = -40 / 20
Z = -2
For the raw scores, we rearrange the Z score formula:
X = Z * σ + μ
(d) For a Z score of 2.4:
X = 2.4 * 20 + 300
X = 48 + 300
X = 348
(e) For a Z score of 1.5:
X = 1.5 * 20 + 300
X = 30 + 300
X = 330
(f) For a Z score of 0:
X = 0 * 20 + 300
X = 0 + 300
X = 300
(g) For a Z score of -4.5:
X = -4.5 * 20 + 300
X = -90 + 300
X = 210
So, the raw scores for the given Z scores are:
(a) 340
(b) 310
(c) 260
(d) 348
(e) 330
(f) 300
(g) 210