Questions: Data Set 4-2 5,3,9,7,6 A raw score of 5 in Data Set 4-2 has a z score of -0.05 -0.5 0.45 -0.45

Transcript text: Data Set 4-2 $5,3,9,7,6$ A raw score of 5 in Data Set 4-2 has a $\boldsymbol{z}$ score of $-0.05$ $-0.5$ 0.45 $-0.45$

Solution

Solution Steps

To find the z-score of a raw score in a data set, we use the formula: $ z = \frac{(X - \mu)}{\sigma} $, where $ X $ is the raw score, $ \mu $ is the mean of the data set, and $ \sigma $ is the standard deviation. First, calculate the mean of the data set. Then, compute the standard deviation. Finally, apply the z-score formula using the raw score of 5.

Step 1: Calculate the Mean

The mean ($\mu$) of the data set $[5, 3, 9, 7, 6]$ is calculated as follows: \[ \mu = \frac{5 + 3 + 9 + 7 + 6}{5} = \frac{30}{5} = 6 \]

Step 2: Calculate the Standard Deviation

The standard deviation ($\sigma$) is calculated using the formula: \[ \sigma = \sqrt{\frac{(5-6)^2 + (3-6)^2 + (9-6)^2 + (7-6)^2 + (6-6)^2}{5}} \] \[ = \sqrt{\frac{1 + 9 + 9 + 1 + 0}{5}} = \sqrt{\frac{20}{5}} = \sqrt{4} = 2 \]

Step 3: Calculate the Z-Score

The z-score ($z$) for the raw score $X = 5$ is calculated using the formula: \[ z = \frac{X - \mu}{\sigma} = \frac{5 - 6}{2} = \frac{-1}{2} = -0.5 \]