Respuesta :
Answer:
(a) Approximately 205 students scored between 540 and 660.
(b) Approximately 287 students scored between 480 and 720.
Step-by-step explanation:
A mound-shaped distribution is a normal distribution since the shape of a normal curve is mound-shaped.
Let X = test score of a student.
It is provided that [tex]X\sim N(\mu = 600, \sigma^{2} = 3600)[/tex].
(a)
The probability of scores between 540 and 660 as follows:
[tex]P(540\leq X\leq 660)=P(\frac{540-600}{\sqrt{3600} }\leq \frac{X-600}{\sqrt{3600} }\leq \frac{660-600}{\sqrt{3600} })\\=P(-1 \leq Z\leq 1)\\= P(Z\leq 1)-P(Z\leq -1)\\=0.8413-0.1587\\=0.6826[/tex]
Use the standard normal table for the probabilities.
The number of students who scored between 540 and 660 is:
300 × 0.6826 = 204.78 ≈ 205
Thus, approximately 205 students scored between 540 and 660.
(b)
The probability of scores between 480 and 720 as follows:
[tex]P(480\leq X\leq 720)=P(\frac{480-600}{\sqrt{3600} }\leq \frac{X-600}{\sqrt{3600} }\leq \frac{720-600}{\sqrt{3600} })\\=P(-2 \leq Z\leq 2)\\= P(Z\leq 2)-P(Z\leq -2)\\=0.9772-0.0228\\=0.9544[/tex]
Use the standard normal table for the probabilities.
The number of students who scored between 480 and 720 is:
300 × 0.9544 = 286.32 ≈ 287
Thus, approximately 287 students scored between 480 and 720.
