Use your understanding of \( z \)-scores to answer the following. Round answers to 2 decimal places (a) Find the \( z \)-score for 28 if the mean of the normal curve is 18 and the standard deviation is 3 . \( z= \) Above the mean Below the mean Equal to the mean What do we know about a value if its \( z \)-score is 0.52 ? (c) Which \( z \)-score corresponds to a value furthest from the mean? 2.76 2.51 1.75

To find the \( z \)-score, we use the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \( X \) is the value (28), \( \mu \) is the mean (18), and \( \sigma \) is the standard deviation (3). So, \[ z = \frac{(28 - 18)}{3} = \frac{10}{3} \approx 3.33 \] The \( z \)-score is approximately 3.33, which is above the mean. If a \( z \)-score is 0.52, it indicates that the value is above the mean, specifically 0.52 standard deviations above it. This means it's closer to the average but still on the higher side. To determine which \( z \)-score corresponds to the value furthest from the mean, we compare the given options: - 2.76 - 2.51 - 1.75 The largest \( z \)-score is 2.76, meaning it is furthest from the mean, regardless of whether it's above or below it.