Respuesta :
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 X represent the scores on a Psychology exam.
The random variable X follows a Normal distribution with mean, μ = 67 and standard deviation, σ = 8.
Assume that the maximum score is 100.
(a)
Compute the probability of the scores that were less than 59% as follows:
[tex]P(X<59)=P(\frac{X-\mu}{\sigma}<\frac{59-67}{8})[/tex]
[tex]=P(Z<-1)\\\\=1-P(Z<1)\\\\=1-0.84134\\\\=0.15866\\\\\approx 0.16[/tex]
*Use a z-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:
[tex]P(X>83)=P(\frac{X-\mu}{\sigma}>\frac{83-67}{8})[/tex]
[tex]=P(Z>2)\\\\=1-P(Z<2)\\\\=1-0.97725\\\\=0.02275\\\\\approx 0.02[/tex]
*Use a z-table.
Thus, the percentage of the scores were over 83% is 2%.
(c)
It is provided n = 160 students took the exam.
Compute the probability of the scores that were over 75% as follows:
[tex]P(X>75)=P(\frac{X-\mu}{\sigma}>\frac{75-67}{8})[/tex]
[tex]=P(Z>1)\\\\=1-P(Z<1)\\\\=1-0.84134\\\\=0.15866\\\\\approx 0.16[/tex]
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:
[tex]\text{Number of Students}=0.16\times 160=25.6\approx 26[/tex]
Thus, the number of students who received a score over 75% is 26.
