The grades on a geometry midterm at Santa Rita are roughly symmetric with [tex]\mu = 82[/tex] and [tex]\sigma = 4.0[/tex]. Emily scored 92 on the exam. Find the z-score for Emily's exam grade. Round to two decimal places.



Answer :

To find the z-score for Emily's exam grade, we need to understand what the z-score represents. The z-score is a measure of how many standard deviations away a particular value is from the mean. It is given by the formula:

[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]

where:
- [tex]\( X \)[/tex] is the value we are interested in,
- [tex]\( \mu \)[/tex] is the mean of the distribution,
- [tex]\( \sigma \)[/tex] is the standard deviation.

For Emily's exam grade, we are given:
- [tex]\( \mu = 82 \)[/tex] (mean score)
- [tex]\( \sigma = 4.0 \)[/tex] (standard deviation)
- Emily's score [tex]\( X = 92 \)[/tex]

Let's substitute these values into the formula:

[tex]\[ z = \frac{92 - 82}{4.0} \][/tex]

Now, calculate the difference in the numerator:

[tex]\[ 92 - 82 = 10 \][/tex]

So the formula becomes:

[tex]\[ z = \frac{10}{4.0} \][/tex]

Next, perform the division:

[tex]\[ z = 2.5 \][/tex]

Therefore, Emily's z-score is:

[tex]\[ z = 2.5 \][/tex]

Since we are asked to round to two decimal places, and the calculated z-score is already at two decimal places, the final z-score is:

[tex]\[ z = 2.5 \][/tex]

So, Emily's z-score for her exam grade is [tex]\( 2.5 \)[/tex].