Question

note: suppose we study the distribution of cumulative grade point average (gpa) data for all students at neumann university. we then discover that this gpa distribution has a bell - shaped histogram with an index of skewness = 0.025. we also discover that the population mean gpa = 2.95, and the standard deviation is: sd = 0.30. please answer the following three (3) questions. #53 what percentage of neumann students would you expect to have a gpa between 2.65 and 3.25? (a) 50% (b) 68% (c) 75% (d) 95% (e) 99.7% #54 what percentage of neumann students would you expect to have a gpa between 2.35 and 3.55? (a) 50% (b) 68% (c) 75% (d) 95% (e) 99.7% #55 what percentage of neumann students would you expect to have a gpa between 2.05 and 3.85? (a) 50% (b) 68% (c) 75% (d) 95% (e) 99.7%

Explanation:

Step1: Calculate z - scores

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $\mu = 2.95$ is the mean and $\sigma=0.30$ is the standard deviation.
For $x = 2.65$, $z_1=\frac{2.65 - 2.95}{0.30}=\frac{- 0.3}{0.3}=-1$.
For $x = 3.25$, $z_2=\frac{3.25 - 2.95}{0.30}=\frac{0.3}{0.3}=1$.
According to the empirical rule of the normal distribution, about 68% of the data lies within 1 standard - deviation of the mean ($z=-1$ to $z = 1$). So for question #53, the percentage of students with GPA between 2.65 and 3.25 is 68%.

Step2: Calculate z - scores for question #54

For $x = 2.35$, $z_1=\frac{2.35 - 2.95}{0.30}=\frac{-0.6}{0.3}=-2$.
For $x = 3.55$, $z_2=\frac{3.55 - 2.95}{0.30}=\frac{0.6}{0.3}=2$.
According to the empirical rule, about 95% of the data lies within 2 standard - deviations of the mean ($z=-2$ to $z = 2$). So for question #54, the percentage of students with GPA between 2.35 and 3.55 is 95%.

Step3: Calculate z - scores for question #55

For $x = 2.05$, $z_1=\frac{2.05 - 2.95}{0.30}=\frac{-0.9}{0.3}=-3$.
For $x = 3.85$, $z_2=\frac{3.85 - 2.95}{0.30}=\frac{0.9}{0.3}=3$.
According to the empirical rule, about 99.7% of the data lies within 3 standard - deviations of the mean ($z=-3$ to $z = 3$). So for question #55, the percentage of students with GPA between 2.05 and 3.85 is 99.7%.

Answer:

#53: B. 68%
#54: D. 95%
#55: E. 99.7%