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].
