Consider the following data set for a population: \( 26,27,32,29,35,38,30,18,31,34 \). This data set has a mean of 30 . Given that the \( z \)-score represents the distance from the mean in terms of the standard deviation, the score in the data set that would have the largest \( z \)-score in magnitude would be

To find the score in the data set that would have the largest \( z \)-score in magnitude, we first need to calculate the standard deviation of the data set. The \( z \)-score is calculated using the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \( X \) is the value from the data set, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. 1. **Calculate the mean (\( \mu \))**: The mean is given as \( 30 \). 2. **Calculate the standard deviation (\( \sigma \))**: First, we find the variance. The variance is calculated using the formula: \[ \sigma^2 = \frac{\sum (X - \mu)^2}{N} \] where \( N \) is the number of data points. The data points are: \( 26, 27, 32, 29, 35, 38, 30, 18, 31, 34 \). Now, we calculate \( (X - \mu)^2 \) for each data point: - For \( 26 \): \( (26 - 30)^2 = (-4)^2 = 16 \) - For \( 27 \): \( (27 - 30)^2 = (-3)^2 = 9 \) - For \( 32 \): \( (32 - 30)^2 = (2)^2 = 4 \) - For \( 29 \): \( (29 - 30)^2 = (-1)^2 = 1 \) - For \( 35 \): \( (35 - 30)^2 = (5)^2 = 25 \) - For \( 38 \): \( (38 - 30)^2 = (8)^2 = 64 \) - For \( 30 \): \( (30 - 30)^2 = (0)^2 = 0 \) - For \( 18 \): \( (18 - 30)^2 = (-12)^2 = 144 \) - For \( 31 \): \( (31 - 30)^2 = (1)^2 = 1 \) - For \( 34 \): \( (34 - 30)^2 = (4)^2 = 16 \) Now, we sum these squared differences: \[ \sum (X - \mu)^2 = 16 + 9 + 4 + 1 + 25 + 64 + 0 + 144 + 1 + 16 = 280 \] Now, we divide by \( N = 10 \) to find the variance: \[ \sigma^2 = \frac{280}{10} = 28 \] Taking the square root gives us the standard deviation: \[ \sigma = \sqrt{28} \approx 5.29 \] 3. **Calculate the \( z \)-scores**: Now we can calculate the \( z \)-scores for each data point: - For \( 26 \): \( z = \frac{(26 - 30)}{5.29} \approx -0.75 \) - For \( 27 \): \( z = \frac{(27 - 30)}{5.29} \approx -0.57 \) - For \( 32 \): \( z = \frac{(32 - 30)}{5.29} \approx 0.38 \) - For \( 29 \): \( z = \frac{(29 - 30)}{5.29} \approx -0.19 \) - For \( 35 \): \( z = \frac{(35 - 30)}{5.29} \approx 0.94 \) - For \( 38 \): \( z = \frac{(38 - 30)}{5.29} \approx 1.51 \) - For \( 30 \): \( z = \frac{(30 - 30)}{5.29} = 0 \) - For \( 18 \): \( z = \frac{(18 - 30)}{5.29} \approx -2.26 \) - For \( 31 \): \( z = \frac{(31 - 30)}{5.29} \approx 0.19 \) - For \( 34 \): \( z = \frac{(34 - 30)}{5.29} \approx 0.75 \) 4. **Identify the largest \( z \)-score in magnitude**: The largest \( z \)-score in magnitude is \( -2.26 \) (for \( 18 \)) and \( 1.51 \) (for \( 38 \)). The largest in absolute value is \( 2.26 \). Thus, the score in the data set that would have the largest \( z \)-score in magnitude is: \[ \boxed{18} \]