Question

Two students, Stephanie and Maria, want to find out who has the higher GPA when compared to each of their schools. Stephanie has a GPA of 3.85, and her school has a mean GPA of 3.1 and a standard deviation of 0.4. Maria has a GPA of 3.8, and her school has a mean of 3.05 and a standard deviation of 0.2. Who has the higher GPA when compared to each of their schools?

Answer 1

Maria has the higher z score, so she has the higher GPA when compared to each of their schools.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

Between Stephanie and Maria, whoever has the higher zscore has the higher GPA when compared to each of their schools.

Stephanie

Stephanie has a GPA of 3.85, and her school has a mean GPA of 3.1 and a standard deviation of 0.4. So we have
`X = 3.85, \mu = 3.1, \sigma = 0.4`. So

`Z = (X - \mu)/(\sigma)`

`Z = (3.85 - 3.1)/(0.4)`

`Z = 1.875`

Maria

Maria has a GPA of 3.8, and her school has a mean of 3.05 and a standard deviation of 0.2. This means that
`X = 3.8, \mu = 3.05, \sigma = 0.2`. So

`Z = (X - \mu)/(\sigma)`

`Z = (3.8 - 3.05)/(0.2)`

`Z = 3.75`

Maria has the higher z score, so she has the higher GPA when compared to each of their schools.

Answer 2

z score for Maria is higher than Stephanie

Explanation:

for Stephanie

GPA = 3.85

Mean of her school GPA = 3.1

Standard deviation = 0.4

`Z =(x -\mu)/(\sigma)`

`= (3.85 -3.1)/(0.4)`

Z =1.875

for Maria

GPA = 3.80

Mean of her school GPA = 3.05

Standard deviation = 0.2

`Z =(x -\mu)/(\sigma)`

`= (3.80 -3.05)/(0.2)`

Z =3.750

therefore z score for Maria is higher than Stephanie