Answer:
a)
b. Joel
c. Juan
d. Linda
b)
b. Joel
c)
a. Jan
f.Susan
d)
a. Jan: 140
b. Joel: 156
c. Juan: 196
d. Linda: 191
e. Robert: 183
f. Susan: 110
Step-by-step explanation:
Z-score:
In a set with mean
and standard deviation
, the z-score of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean, positive z-scores are above the mean, negative are below the mean and 0 is the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Question a:
Robert, Juan and Linda had positive z-scores, so they scored above the mean, and the correct options are c,d,e.
(b) Which of these students scored on the mean?
Joel, which had a z-score of 0, so the correct option is b.
(c) Which of these students scored below the mean?
Jan and Susan had negative z-scores, so them, options a and f.
Question d:
We have that
, so we have to find X for each student.
Jan:
Z = -0.65. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-0.65 = \frac{X - 156}{24}](https://tex.z-dn.net/?f=-0.65%20%3D%20%5Cfrac%7BX%20-%20156%7D%7B24%7D)
![X - 156 = -0.65*24](https://tex.z-dn.net/?f=X%20-%20156%20%3D%20-0.65%2A24)
![X = 140](https://tex.z-dn.net/?f=X%20%3D%20140)
b. Joel
Z = 0, so:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![0 = \frac{X - 156}{24}](https://tex.z-dn.net/?f=0%20%3D%20%5Cfrac%7BX%20-%20156%7D%7B24%7D)
![X - 156 = 0*24](https://tex.z-dn.net/?f=X%20-%20156%20%3D%200%2A24)
![X = 156](https://tex.z-dn.net/?f=X%20%3D%20156)
c. Juan
Z = 1.66, so:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.66 = \frac{X - 156}{24}](https://tex.z-dn.net/?f=1.66%20%3D%20%5Cfrac%7BX%20-%20156%7D%7B24%7D)
![X - 156 = 1.66*24](https://tex.z-dn.net/?f=X%20-%20156%20%3D%201.66%2A24)
![X = 196](https://tex.z-dn.net/?f=X%20%3D%20196)
d. Linda
Z = 1.46. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.46 = \frac{X - 156}{24}](https://tex.z-dn.net/?f=1.46%20%3D%20%5Cfrac%7BX%20-%20156%7D%7B24%7D)
![X - 156 = 1.46*24](https://tex.z-dn.net/?f=X%20-%20156%20%3D%201.46%2A24)
![X = 191](https://tex.z-dn.net/?f=X%20%3D%20191)
e. Robert
Z = 1.11. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.11 = \frac{X - 156}{24}](https://tex.z-dn.net/?f=1.11%20%3D%20%5Cfrac%7BX%20-%20156%7D%7B24%7D)
![X - 156 = 1.11*24](https://tex.z-dn.net/?f=X%20-%20156%20%3D%201.11%2A24)
![X = 183](https://tex.z-dn.net/?f=X%20%3D%20183)
f. Susan
Z = -1.9. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-1.9 = \frac{X - 156}{24}](https://tex.z-dn.net/?f=-1.9%20%3D%20%5Cfrac%7BX%20-%20156%7D%7B24%7D)
![X - 156 = -1.9*24](https://tex.z-dn.net/?f=X%20-%20156%20%3D%20-1.9%2A24)
![X = 110](https://tex.z-dn.net/?f=X%20%3D%20110)