14. Find the \( z \)-score corresponding to \( X=24 \) and the \( X \) value corresponding to \( z=+1.50 \) for each of the following samples. \( \begin{array}{l}\text { a. } M=20 \text { and } s=12 \\ \text { b. } M=20 \text { and } s=4 \\ \text { c. } M=30 \text { and } s=8 \\ \text { d. } M=30 \text { and } s=10\end{array} \)

To find the \( z \)-score corresponding to \( X=24 \) and the \( X \) value corresponding to \( z=+1.50 \), we will use the formula for calculating \( z \)-scores and \( X \) values. The formula for calculating the \( z \)-score is: \[ z = \frac{X - M}{s} \] And the formula to find \( X \) when you know \( z \) is: \[ X = M + z \times s \] Now, let’s apply these formulas to the different samples. **a. M=20 and s=12:** - For \( X=24 \): \[ z = \frac{24 - 20}{12} = \frac{4}{12} \approx 0.33 \] - For \( z=+1.5 \): \[ X = 20 + 1.5 \times 12 = 20 + 18 = 38 \] **b. M=20 and s=4:** - For \( X=24 \): \[ z = \frac{24 - 20}{4} = \frac{4}{4} = 1 \] - For \( z=+1.5 \): \[ X = 20 + 1.5 \times 4 = 20 + 6 = 26 \] **c. M=30 and s=8:** - For \( X=24 \): \[ z = \frac{24 - 30}{8} = \frac{-6}{8} = -0.75 \] - For \( z=+1.5 \): \[ X = 30 + 1.5 \times 8 = 30 + 12 = 42 \] **d. M=30 and s=10:** - For \( X=24 \): \[ z = \frac{24 - 30}{10} = \frac{-6}{10} = -0.6 \] - For \( z=+1.5 \): \[ X = 30 + 1.5 \times 10 = 30 + 15 = 45 \] Here’s the summary of your calculations: - a: \( z \approx 0.33 \), \( X = 38 \) - b: \( z = 1 \), \( X = 26 \) - c: \( z = -0.75 \), \( X = 42 \) - d: \( z = -0.6 \), \( X = 45 \)