Question

question 2 of 15
rebecca and her father both graduated from the same high school. when her father looked at rebeccas transcript, he noticed that her high school gpa (4.2) was higher than his high school gpa (3.9). after letting rebecca gloat for a minute, he pointed out that there were no weighted grades when he went to school. to settle their argument, they called the registrar at the school and got information about the distribution of gpa in each of their graduation years. when the father graduated, the mean gpa was 2.8 with a standard deviation of 0.6. when rebecca graduated, the mean gpa was 3.2 with a standard deviation of 0.7.
(a) calculate the standardized score (z - score) for rebecca and her father.
rebecca: $z=square$ (round to 2 decimal places.)
rebeccas father: $z=square$ (round to 2 decimal places.)
(b) who had the better gpa, relatively speaking? explain your reasoning.
rebeccas father had a $square$ gpa relative to his peers, because his z - score was $square$ rebeccas.
rebeccas father had a gpa that was $square$ of his class, while rebecca had a gpa that was only $square$ of her class.

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the individual value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Calculate Rebecca's z - score

Rebecca's GPA $x = 4.2$, mean $\mu=3.2$, and standard deviation $\sigma = 0.7$. Substitute into the formula: $z=\frac{4.2 - 3.2}{0.7}=\frac{1}{0.7}\approx1.43$.

Step3: Calculate Rebecca's father's z - score

Rebecca's father's GPA $x = 3.9$, mean $\mu = 2.8$, and standard deviation $\sigma=0.6$. Substitute into the formula: $z=\frac{3.9 - 2.8}{0.6}=\frac{1.1}{0.6}\approx1.83$.

Answer:

(a)
Rebecca: $z\approx1.43$
Rebecca's father: $z\approx1.83$
(b)
Rebecca's father had a better GPA relative to his peers, because his z - score was higher than Rebecca's. Rebecca's father had a GPA that was in the upper part of his class, while Rebecca had a GPA that was only above - average of her class.