In a statistics class, the mean and standard deviation of the scores obtained by students on a test were found to be 73 and 20 , respectively. If the scores are normally distributed, what percentage of students scored below 78 ?
To determine the percentage of students who scored below 78 , we need to calculate the z-score corresponding to that value and then find the area under the normal curve to the left of that z-score.
The z-score is calculated using the formula:
where:
is the value we want to find the percentage below (in this case, 78 ),
is the mean (73), and
is the standard deviation (20).
Plugging in the values:
Now we need to find the area to the left of this z-score in the standard normal distribution table (also known as the z-table) or by using a statistical software. The area represents the percentage of students who scored below 78 .
Looking up the z-score of 0.25 in the z-table, we find that the area to the left is approximately 0.5987 .
So, approximately 59.87% of the students scored below 78 on the test.
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