Question 19 A sample of scores from a research study examining stress in the workplace has a mean of 7 and a standard deviation of 2.3. John has scored a 6.2 in his stress survey. What is John's \( Z \) score for the survey? -0.31

To find John's \( Z \) score for the survey, we can use the formula for calculating the \( Z \) score: \[ Z = \frac{X - \mu}{\sigma} \] where: - \( X \) is John's score (6.2), - \( \mu \) is the mean of the scores (7), - \( \sigma \) is the standard deviation of the scores (2.3). Substitute the given values into the formula to find John's \( Z \) score. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(6.2-7\right)}{2.3}\) - step1: Remove the parentheses: \(\frac{6.2-7}{2.3}\) - step2: Subtract the numbers: \(\frac{-0.8}{2.3}\) - step3: Convert the expressions: \(\frac{-\frac{4}{5}}{\frac{23}{10}}\) - step4: Multiply by the reciprocal: \(-\frac{4}{5}\times \frac{10}{23}\) - step5: Reduce the numbers: \(-4\times \frac{2}{23}\) - step6: Multiply: \(-\frac{4\times 2}{23}\) - step7: Multiply: \(-\frac{8}{23}\) John's \( Z \) score for the survey is approximately -0.3478. This means that John's score is approximately 0.3478 standard deviations below the mean.