Question

In a test administration to 1000 student the average score is 42 and standard deviation is 24,
1) the number of student exceeding a score of 50.
2) the number of lying between 30 and 54.
3) the value of score exceed by the top 100 student.

Answer 1

The number of students exceeding a score of 50 is 370.

The number of students whose score lie between 30 and 54 is 383.

The value of score exceeds by the top 100 student 73.

Given: 1000 students took a test where the average score is 42 and standard deviation is 24.

To find: The number of students exceeding a score of 50.

The number of students whose score lie between 30 and 54.

The value of score exceeds by the top 100 student.

Solution:

Total number of students, n = 1000

Let us assume that the given test scores follow normal distribution.

(1) Number of students with score more than 50,

P(x > 50)

P(x > 50) = P(Z > (50–42)/24)

= P(Z > 8/24)

= P(Z > 0.3333)

= 1.0 - P(Z< 0.3333)

Now, referring the Z table, we get 1.0 - (0.5 + 0.1305) = 0.3695.

So, the number of students who exceed the score 50 are - 1000*0.3695 = 369.5 or 370

Approximately 370 students exceeded the score 50.

(2) P ( 30 < X < 54 )

As we know, Z = (x-μ)/σ

For score between 30 and 54, there is  z of -0.5 and +0.5,

which is a probability of 0.3829 or

383 students (that is 12 points on either side of the mean and 12/24 is 1/2, dividing by the standard deviation)

The number of students whose score lie between 30 and 54 is 383.

(3) The value of score exceed by the top 100 students.

Top 100 students is at the 90th percentile

which means, z = + 1.28

using the formula above z * standard deviation = 30.72

That is,

x-mean

so, x = mean + 30.72 or

Score -  72.72 which is 73 to the nearest integer.

The value of score exceed by the top 100 students is 73.

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