A set of data items is normally distributed with a mean of 23 and a standard deviation of 5.8 . Find the data value in the distribution that corresponds to each of the following \( z \)-scores. Round your answers to the nearest tenth. (a) \( z=-1.2 \) (b) \( z=1.43 \)

To find the data value \( x \) corresponding to a given \( z \)-score in a normally distributed dataset, you can use the following formula: \[ x = \mu + z \cdot \sigma \] where: - \( \mu \) is the mean of the distribution. - \( \sigma \) is the standard deviation. - \( z \) is the z-score. Given: - Mean (\( \mu \)) = 23 - Standard deviation (\( \sigma \)) = 5.8 Let's calculate the data values for each \( z \)-score. ### (a) \( z = -1.2 \) \[ x = 23 + (-1.2) \cdot 5.8 \] \[ x = 23 - (1.2 \times 5.8) = 23 - 6.96 = 16.04 \] Rounded to the nearest tenth: \[ x \approx 16.0 \] ### (b) \( z = 1.43 \) \[ x = 23 + 1.43 \cdot 5.8 \] \[ x = 23 + (1.43 \times 5.8) = 23 + 8.294 = 31.294 \] Rounded to the nearest tenth: \[ x \approx 31.3 \] **Final Answers:** - **(a)** The data value is **16.0**. - **(b)** The data value is **31.3**.