Solve these problems 4-5. Find the probability that a piece of data randomly selected from a normal population will have a standard (2) score- \( \begin{array}{ll}\text { a. greater than }-1.5 & \text { b. Less than } 2.13\end{array} \)

Sure, let's solve each part of the problem step by step. We'll use the standard normal distribution (z-distribution) to determine the probabilities. ### Problem 4-5 **Find the probability that a piece of data randomly selected from a normal population will have a standard (z) score:** #### a. **Greater than -1.5** 1. **Understand the Problem:** - We need to find \( P(Z > -1.5) \), where \( Z \) is the standard normal variable. 2. **Use the Standard Normal Distribution Table:** - The table typically provides \( P(Z < z) \). - First, find \( P(Z < -1.5) \). 3. **Find \( P(Z < -1.5) \):** - From the standard normal table, \( P(Z < -1.5) = 0.0668 \). 4. **Calculate \( P(Z > -1.5) \):** \[ P(Z > -1.5) = 1 - P(Z < -1.5) = 1 - 0.0668 = 0.9332 \] 5. **Conclusion:** - The probability that a randomly selected data point has a z-score greater than -1.5 is **0.9332** or **93.32%**. #### b. **Less than 2.13** 1. **Understand the Problem:** - We need to find \( P(Z < 2.13) \). 2. **Use the Standard Normal Distribution Table:** - Find \( P(Z < 2.13) \) directly from the table. 3. **Find \( P(Z < 2.13) \):** - From the standard normal table, \( P(Z < 2.13) \approx 0.9834 \). 4. **Conclusion:** - The probability that a randomly selected data point has a z-score less than 2.13 is **0.9834** or **98.34%**. ### Summary of Answers: - **a.** \( P(Z > -1.5) = 0.9332 \) (93.32%) - **b.** \( P(Z < 2.13) = 0.9834 \) (98.34%)