Question

1. a student earned 82 on a test where the mean was 76 and the standard deviation was 4.

a. find the z - score.
b. explain what the z - score means about the students performance.

Explanation:

Part a

Step1: Recall z - score formula

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

Step2: Identify values

We know that $x = 82$, $\mu=76$, and $\sigma = 4$.

Step3: Substitute values into formula

Substitute the values into the formula: $z=\frac{82 - 76}{4}$.

Step4: Calculate numerator and denominator

First, calculate the numerator: $82-76 = 6$. Then, divide by the denominator: $\frac{6}{4}=1.5$.

A z - score of $1.5$ means that the student's test score of 82 is $1.5$ standard deviations above the mean score of 76. In a normal distribution (if we assume the test scores are normally distributed), a positive z - score indicates that the score is above the mean. A z - score of $1.5$ shows that the student performed better than the average student (since the mean represents the average performance), and the magnitude of the z - score tells us how far above the mean the score is in terms of standard deviations.

Answer:

The z - score is $1.5$.

Part b