Answer:
To solve for the value of "k" in each scenario, you can use the standard normal distribution table (z-table) or a calculator. Here are the steps to find the values of "k" for each case:
A. For \( P(X > k) = 0.4325 \):
- Find the z-score corresponding to \( 1 - 0.4325 = 0.5675 \) in the z-table.
- The z-score is approximately 0.18.
- Use the z-score formula: \( z = \frac{x - \mu}{\sigma} \) to solve for "x" (the value of "k").
- Plug in the given values: \( 0.18 = \frac{k - 32.5}{3.8} \) and solve for "k".
B. For \( P(X \leq k) = 0.854 \):
- Find the z-score corresponding to \( 0.854 \) in the z-table.
- The z-score is approximately 1.04.
- Use the z-score formula: \( z = \frac{x - \mu}{\sigma} \) to solve for "x" (the value of "k").
- Plug in the given values: \( 1.04 = \frac{k - 32.5}{3.8} \) and solve for "k".
C. For \( P(30 < X < k) = 0.5635 \):
- Find the z-scores corresponding to \( 0.5 - \frac{0.5635}{2} = 0.21725 \) and \( 0.5 + \frac{0.5635}{2} = 0.78275 \) in the z-table.
- The z-scores are approximately -0.78 and 0.69.
- Use the z-score formula: \( z = \frac{x - \mu}{\sigma} \) to solve for "x" (the values of "k" for both ends).
- Plug in the given values and solve for "k".
Remember to solve for "k" in each case and then round your answer to 3 decimal places.