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% #56 what is the z - score which corresponds to the gpa = 2.35? (a) 0.00 (b) 1.00 (c) 2.00 (d) - 1.00 (e) - 2.00 #57 what is the z - score which corresponds to the gpa = 2.65? (a) 0.00 (b) 1.00 (c) 2.00 (d) - 1.00 (e) - 2.00 #58 what is the z - score which corresponds to the gpa = 2.95? (a) 0.00 (b) 1.00 (c) 2.00 (d) - 1.00 (e) - 2.00

Explanation:

Step1: Recall the formula for the z - score

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the population mean, and $\sigma$ is the standard deviation. Here, $\mu = 2.95$ and $\sigma=0.30$.

Step2: Recall the empirical rule for normal distributions

• Approximately 68% of the data lies within 1 standard - deviation of the mean ($z=\pm1$), i.e., $\mu\pm\sigma$.
• Approximately 95% of the data lies within 2 standard - deviations of the mean ($z = \pm2$), i.e., $\mu\pm2\sigma$.
• Approximately 99.7% of the data lies within 3 standard - deviations of the mean ($z=\pm3$), i.e., $\mu\pm3\sigma$.

For #53

1. Calculate the z - scores:

• For $x = 2.65$, $z_1=\frac{2.65 - 2.95}{0.30}=\frac{- 0.30}{0.30}=-1$.
• For $x = 3.25$, $z_2=\frac{3.25 - 2.95}{0.30}=\frac{0.30}{0.30}=1$.
• Since the values are within 1 standard - deviation of the mean, the percentage of students with GPA between 2.65 and 3.25 is 68%.

For #54

1. Calculate the z - scores:

• For $x = 2.35$, $z_1=\frac{2.35 - 2.95}{0.30}=\frac{-0.60}{0.30}=-2$.
• For $x = 3.55$, $z_2=\frac{3.55 - 2.95}{0.30}=\frac{0.60}{0.30}=2$.
• Since the values are within 2 standard - deviations of the mean, the percentage of students with GPA between 2.35 and 3.55 is 95%.

For #55

1. Calculate the z - scores:

• For $x = 2.05$, $z_1=\frac{2.05 - 2.95}{0.30}=\frac{-0.90}{0.30}=-3$.
• For $x = 3.85$, $z_2=\frac{3.85 - 2.95}{0.30}=\frac{0.90}{0.30}=3$.
• Since the values are within 3 standard - deviations of the mean, the percentage of students with GPA between 2.05 and 3.85 is 99.7%.

For #56

1. Calculate the z - score:

• Using the formula $z=\frac{x-\mu}{\sigma}$, with $x = 2.35$, $\mu = 2.95$, and $\sigma = 0.30$, we have $z=\frac{2.35 - 2.95}{0.30}=\frac{-0.60}{0.30}=-2$.

For #57

1. Calculate the z - score:

• Using the formula $z=\frac{x-\mu}{\sigma}$, with $x = 2.65$, $\mu = 2.95$, and $\sigma = 0.30$, we have $z=\frac{2.65 - 2.95}{0.30}=\frac{-0.30}{0.30}=-1$.

For #58

1. Calculate the z - score:

• Using the formula $z=\frac{x-\mu}{\sigma}$, with $x = 2.95$, $\mu = 2.95$, and $\sigma = 0.30$, we have $z=\frac{2.95 - 2.95}{0.30}=0$.

Answer:

#53: B. 68%
#54: D. 95%
#55: E. 99.7%
#56: E. - 2.00
#57: D. - 1.00
#58: A. 0.00