Grade point average

The average GPA is 2.78, with a standard deviation of 0.45. If GPA is normally distributed, what percentage of the students have the following GPAs? Solve for the z-score and report the appropriate percentage:

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

What students at the bottom of the 20% have what GPA?

Correct answer:

a =  14.31 %
b =  4.15 %
c =  95.85 %
d =  31.25 %
e =  67.83 %

Step-by-step explanation:

$$\begin{gathered} \mu =2.78 \\ \sigma =0.45 \\ a=N(\mu ,\sigma )<2.3 \\ {z}_1=\frac{2.3-\mu }{\sigma }=\frac{2.3-2.78}{0.45}=-\frac{16}{15}\doteq -1.0667 \\ a=100\cdot 0.1431=14.31\% \end{gathered}$$

Use the normal distribution table.

$$\begin{gathered} z_2=\frac{2.0-\mu }{\sigma }=\frac{2.0-2.78}{0.45}=-\frac{26}{15}\doteq -1.7333 \\ b=100\cdot 0.0415=4.15\% \end{gathered}$$

Use the normal distribution table.

$c=100-b=100-4.15=95.85\%$

Use the normal distribution table.

$d=100\cdot 0.3125=31.25\%$

Use the normal distribution table.

$e=100\cdot 0.6783=67.83\%$

Use the normal distribution table.

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