A set of final examination grades in an introductory statist

A set of final examination grades in an introductory statistics course was found to be normally distributed with a mean of 75 and a standard deviation of 7.

What is the probability that a student scored between 68 and 90?

Solution

To calculate probabilities from a normal distribution we need to standardise the given values and find the area between the given values using the z-tables. Z- tables are based on the known facts that;

1. the total area under it=1

2. the standard deviation of the distribution=1

and

3. Its mean = 0

The z- standardisation formula for any given X-value= X - Mean/ Standard Deviation

therefore 68_z = 68-75/7= -1

and

90_z= 90-75/7= 2.14

From the z tables the area between 0 and -1=0.3413

and that between 0 and 2.14=0.4838

Therefore the probability that a student scored between 68 and 90= 0.3413 + 0.4383 = 0.7796 or 77.96%