A sample of university students has an average GPA of 2.78 with a standard deviation of 0.45. If GPA is normally distributed, what percentage of the students has GPAs Z score Area.
a. less than 2.30
b. less than 2.00
c. more than 2.00
d. more than 3.00
e. between 2.50 and 3.50
f. between 2.00 and 2.50

Question

A sample of university students has an average GPA of 2.78 with a standard deviation of 0.45. If GPA is normally distributed, what percentage of the students has GPAs Z score Area. 
a. less than 2.30 
b. less than 2.00 
c. more than 2.00 
d. more than 3.00 
e. between 2.50 and 3.50 
f. between 2.00 and 2.50

Quizplus · Accepted Answer

To answer these questions, we need to calculate the Z-scores for the given GPAs and then use the standard normal distribution to find the corresponding areas (percentages of students). The Z-score is calculated using the formula:
$$ Z = \frac{(X - \mu)}{\sigma} $$
where $ X $ is the value of the GPA, $ \mu $ is the mean GPA, and $ \sigma $ is the standard deviation.
Given:
- Mean GPA ($ \mu $) = 2.78
- Standard deviation ($ \sigma $) = 0.45
Let's calculate the Z-scores and the corresponding percentages for each case:
a. less than 2.30
$$ Z = \frac{(2.30 - 2.78)}{0.45} = \frac{-0.48}{0.45} \approx -1.067 $$
Using a Z-table or a standard normal distribution calculator, we find the area to the left of Z = -1.067. This area represents the percentage of students with a GPA less than 2.30.
b. less than 2.00
$$ Z = \frac{(2.00 - 2.78)}{0.45} = \frac{-0.78}{0.45} \approx -1.733 $$
Find the area to the left of Z = -1.733.
c. more than 2.00
This is the complement of part b. We can find the area to the right of Z = -1.733, which is 1 minus the area to the left of Z = -1.733.
d. more than 3.00
$$ Z = \frac{(3.00 - 2.78)}{0.45} = \frac{0.22}{0.45} \approx 0.489 $$
Find the area to the right of Z = 0.489.
e. between 2.50 and 3.50
Calculate the Z-scores for both GPAs:
For 2.50:
$$ Z = \frac{(2.50 - 2.78)}{0.45} = \frac{-0.28}{0.45} \approx -0.622 $$
For 3.50:
$$ Z = \frac{(3.50 - 2.78)}{0.45} = \frac{0.72}{0.45} \approx 1.600 $$
Find the area between Z = -0.622 and Z = 1.600.
f. between 2.00 and 2.50
We already have the Z-score for 2.00 from part b and for 2.50 from part e. Find the area between Z = -1.733 and Z = -0.622.
To find the exact percentages, you would typically use a Z-table or statistical software. For the purposes of this explanation, I will not provide the exact percentages, but you can easily find them using the methods described above.

A Sample of University Students Has an Average GPA of 2.78