There are 75 students in a class and their average marks is 50 and S.D of marks is 5. Number of studentswho have secured more than 60 marks (Given that area under the normal curve for z = 2 is 0.4772)?

Question

Answer

Calculation of Number of Students with More than 60 Marks

Given, total number of students (n) = 75

Average marks (μ) = 50

S.D of marks (σ) = 5

Formula for z-score = (X - μ) / σ

For X = 60, z-score = (60 - 50) / 5 = 2

Area under normal curve for z = 2 is 0.4772 (given)

Formula for converting z-score to percentage = area under curve × 100

Therefore, percentage of students who scored more than 60 = (1 - 0.4772) × 100 = 52.28%

Number of students who scored more than 60 = percentage × total number of students = 52.28% × 75 = 39.21 ≈ 39 students

Explanation

To calculate the number of students who scored more than 60 marks, we first need to find the z-score for 60 marks using the formula (X - μ) / σ. Once we have the z-score, we can use the standard normal distribution table or calculator to find the area under the normal curve for that z-score. In this case, the area under the curve for z = 2 is given as 0.4772. This means that 47.72% of the students scored more than 60 marks.

However, we need to find the actual number of students who scored more than 60 marks. To do this, we need to convert the percentage to a number using the formula percentage × total number of students. The result is rounded off to the nearest whole number, which gives us 39 students.

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