At one college, GPA's are normally distributed with a mean of 2.7 and a standard deviation of 0.5. What percentage of students at the college have a GPA between 2.2 and 3.2?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the two Z scores.
This is simple enough that the approximation could be memorized.
68
68%
To find the percentage of students at the college with a GPA between 2.2 and 3.2, we need to calculate the area under the normal distribution curve between these two values.
First, let's standardize the values by using the z-score formula:
z1 = (2.2 - 2.7) / 0.5
z2 = (3.2 - 2.7) / 0.5
Calculating the z-scores:
z1 = -1
z2 = 1
Next, we can use a standard normal distribution table or a statistical calculator to find the proportion (percentage) of values between these z-scores.
Looking up the values in the standard normal distribution table, we find:
Pr( z < -1 ) = 0.1587 (area to the left of z1)
Pr( z < 1 ) = 0.8413 (area to the left of z2)
To find the percentage between these two values, we subtract the lower percentage from the higher percentage:
Pr( -1 < z < 1 ) = Pr( z < 1 ) - Pr( z < -1 )
= 0.8413 - 0.1587
= 0.6826
Therefore, approximately 68.26% of students at the college have a GPA between 2.2 and 3.2.