10,000 students took a math test. If there average score was 40 with standard deviation of 10, how many students must have scored between 20 and 40 marks?
3,772
2,227
5,000
3,334
4,772
20 is 2 std's out
2 std's is 47.7%
So, I'd pick 4772
To find the number of students who scored between 20 and 40 marks, we can use the concept of the standard normal distribution.
Step 1: Convert the scores to a standard normal distribution.
To do this, we need to calculate the z-scores for each score within the distribution. The z-score measures the number of standard deviations a particular value is from the mean.
The formula to calculate the z-score is:
z = (x - mean) / standard deviation
For the lower score of 20:
z1 = (20 - 40) / 10
z1 = -20 / 10
z1 = -2
For the upper score of 40:
z2 = (40 - 40) / 10
z2 = 0 / 10
z2 = 0
Step 2: Find the area under the normal curve between the z-scores of -2 and 0.
This represents the proportion of students who scored between 20 and 40 in the standard normal distribution.
We can use a standard normal distribution table or a statistical software to find this area. Using a standard normal distribution table, the area between z = -2 and z = 0 is 0.4772.
Step 3: Convert the proportion back to the actual number of students.
Since there were 10,000 students in total, we can multiply the proportion by 10,000 to find the number of students who scored between 20 and 40.
Number of students = proportion * total number of students
Number of students = 0.4772 * 10,000
Number of students ≈ 4,772
Therefore, the answer is 4,772.