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 <em>X</em> = test score of a student.
It is provided that
.
(a)
The probability of scores between 540 and 660 as follows:
![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](https://tex.z-dn.net/?f=P%28540%5Cleq%20X%5Cleq%20660%29%3DP%28%5Cfrac%7B540-600%7D%7B%5Csqrt%7B3600%7D%20%7D%5Cleq%20%5Cfrac%7BX-600%7D%7B%5Csqrt%7B3600%7D%20%7D%5Cleq%20%5Cfrac%7B660-600%7D%7B%5Csqrt%7B3600%7D%20%7D%29%5C%5C%3DP%28-1%20%5Cleq%20Z%5Cleq%201%29%5C%5C%3D%20P%28Z%5Cleq%201%29-P%28Z%5Cleq%20-1%29%5C%5C%3D0.8413-0.1587%5C%5C%3D0.6826)
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:
![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](https://tex.z-dn.net/?f=P%28480%5Cleq%20X%5Cleq%20720%29%3DP%28%5Cfrac%7B480-600%7D%7B%5Csqrt%7B3600%7D%20%7D%5Cleq%20%5Cfrac%7BX-600%7D%7B%5Csqrt%7B3600%7D%20%7D%5Cleq%20%5Cfrac%7B720-600%7D%7B%5Csqrt%7B3600%7D%20%7D%29%5C%5C%3DP%28-2%20%5Cleq%20Z%5Cleq%202%29%5C%5C%3D%20P%28Z%5Cleq%202%29-P%28Z%5Cleq%20-2%29%5C%5C%3D0.9772-0.0228%5C%5C%3D0.9544)
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.