the mean and standard deviation of the grades of 500 students who took an maths exam were 69 and 7 respectively. if the grades have a bell shaped distribution, how many of these students received a grade of more than 83 marks?
To find the number of students who received a grade of more than 83 marks, we need to use the concept of z-scores.
A z-score measures the number of standard deviations a data point is from the mean. It can be calculated using the formula:
z = (X - μ) / σ
Where:
- X is the value (grade) we want to find the z-score for,
- μ is the mean of the distribution, and
- σ is the standard deviation of the distribution.
In this case, the mean (μ) is 69 and the standard deviation (σ) is 7.
To find the z-score for a grade of 83:
z = (83 - 69) / 7
z = 14 / 7
z = 2
Since the distribution is bell-shaped (normally distributed), we can use a z-table to find the percentage of students who scored above 83 marks.
In the z-table, a z-score of 2 corresponds to a percentage of approximately 97.7%. This means that approximately 97.7% of the students scored below 83 marks.
To find the number of students who scored more than 83 marks, we subtract this percentage from 100% (total number of students):
Number of students = Total number of students * (1 - Percentage below 83 marks)
Given that there are 500 students:
Number of students = 500 * (1 - 0.977)
Number of students = 500 * 0.023
Number of students ≈ 11.5
Since we can't have partial students, we round the result to the nearest whole number. Therefore, approximately 12 students received a grade of more than 83 marks.