Answer:
(a) The percentage of the scores were less than 59% is 16%.
(b) The percentage of the scores were over 83% is 2%.
(c) The number of students who received a score over 75% is 26.
Step-by-step explanation:
Let the random variable <em>X</em> represent the scores on a Psychology exam.
The random variable <em>X</em> follows a Normal distribution with mean, <em>μ</em> = 67 and standard deviation, <em>σ</em> = 8.
Assume that the maximum score is 100.
(a)
Compute the probability of the scores that were less than 59% as follows:
![P(X](https://tex.z-dn.net/?f=P%28X%3C59%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B59-67%7D%7B8%7D%29)
![=P(Z](https://tex.z-dn.net/?f=%3DP%28Z%3C-1%29%5C%5C%5C%5C%3D1-P%28Z%3C1%29%5C%5C%5C%5C%3D1-0.84134%5C%5C%5C%5C%3D0.15866%5C%5C%5C%5C%5Capprox%200.16)
*Use a <em>z</em>-table.
Thus, the percentage of the scores were less than 59% is 16%.
(b)
Compute the probability of the scores that were over 83% as follows:
![P(X>83)=P(\frac{X-\mu}{\sigma}>\frac{83-67}{8})](https://tex.z-dn.net/?f=P%28X%3E83%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B83-67%7D%7B8%7D%29)
![=P(Z>2)\\\\=1-P(Z](https://tex.z-dn.net/?f=%3DP%28Z%3E2%29%5C%5C%5C%5C%3D1-P%28Z%3C2%29%5C%5C%5C%5C%3D1-0.97725%5C%5C%5C%5C%3D0.02275%5C%5C%5C%5C%5Capprox%200.02)
*Use a <em>z</em>-table.
Thus, the percentage of the scores were over 83% is 2%.
(c)
It is provided <em>n</em> = 160 students took the exam.
Compute the probability of the scores that were over 75% as follows:
![P(X>75)=P(\frac{X-\mu}{\sigma}>\frac{75-67}{8})](https://tex.z-dn.net/?f=P%28X%3E75%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B75-67%7D%7B8%7D%29)
![=P(Z>1)\\\\=1-P(Z](https://tex.z-dn.net/?f=%3DP%28Z%3E1%29%5C%5C%5C%5C%3D1-P%28Z%3C1%29%5C%5C%5C%5C%3D1-0.84134%5C%5C%5C%5C%3D0.15866%5C%5C%5C%5C%5Capprox%200.16)
The percentage of students who received a score over 75% is 16%.
Compute the number of students who received a score over 75% as follows:
![\text{Number of Students}=0.16\times 160=25.6\approx 26](https://tex.z-dn.net/?f=%5Ctext%7BNumber%20of%20Students%7D%3D0.16%5Ctimes%20160%3D25.6%5Capprox%2026)
Thus, the number of students who received a score over 75% is 26.