Question

1. the average height of students is 65 inches with a standard deviation of 3 inches.

a. find the z - score for a student who is 70 inches tall.
b. interpret the result.

Explanation:

Part (a)

Step1: Recall z - score formula

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

Step2: Identify values

We are given that $\mu = 65$ (the average height), $\sigma=3$ (the standard deviation), and $x = 70$ (the height of the student).

Step3: Substitute values into the formula

Substitute $x = 70$, $\mu=65$, and $\sigma = 3$ into the z - score formula:
$z=\frac{70 - 65}{3}$

Step4: Calculate the result

First, calculate the numerator: $70-65 = 5$. Then divide by the denominator: $\frac{5}{3}\approx1.67$ (rounded to two decimal places).

The z - score represents the number of standard deviations a data point is from the mean. A z - score of approximately 1.67 means that the height of the student (70 inches) is about 1.67 standard deviations above the mean height of the students. Since the z - score is positive, it is above the mean.

Answer:

The z - score is $\frac{5}{3}\approx1.67$

Part (b)