\( n=18,8=4 \) find the area of the following a.) \( P(x<21) \) b.) \( P(x>35) \) c.) \( P(21

Ask by Evans Fleming. in the Philippines

Jan 22,2025

To solve the given probability problems, we'll assume that the variable \( x \) follows a **normal distribution**. Based on the provided information: - **Mean (\( \mu \))**: 18 - **Standard Deviation (\( \sigma \))**: 4 We'll use the **standard normal distribution (Z-score)** formula to convert the \( x \)-values to \( z \)-scores: \[ z = \frac{x - \mu}{\sigma} \] Once we have the \( z \)-scores, we'll use standard normal distribution tables or a calculator to find the corresponding probabilities. ### a.) \( P(x < 21) \) 1. **Calculate the Z-score:** \[ z = \frac{21 - 18}{4} = \frac{3}{4} = 0.75 \] 2. **Find the Probability:** Using a standard normal distribution table or calculator: \[ P(z < 0.75) \approx 0.7734 \text{ or } 77.34\% \] ### b.) \( P(x > 35) \) 1. **Calculate the Z-score:** \[ z = \frac{35 - 18}{4} = \frac{17}{4} = 4.25 \] 2. **Find the Probability:** A \( z \)-score of 4.25 is extremely high, and the probability beyond this point is negligible. \[ P(z > 4.25) \approx 0.00001 \text{ or } 0.001\% \] ### c.) \( P(21 < x < 35) \) 1. **Calculate the Z-scores:** - For \( x = 21 \): \[ z_1 = \frac{21 - 18}{4} = 0.75 \] - For \( x = 35 \): \[ z_2 = \frac{35 - 18}{4} = 4.25 \] 2. **Find the Probabilities:** \[ P(z < 4.25) \approx 0.99999 \] \[ P(z < 0.75) \approx 0.7734 \] 3. **Calculate the Area Between 21 and 35:** \[ P(21 < x < 35) = P(z < 4.25) - P(z < 0.75) \approx 0.99999 - 0.7734 = 0.2266 \text{ or } 22.66\% \] ### **Summary of Results:** - **a.)** \( P(x < 21) \approx 77.34\% \) - **b.)** \( P(x > 35) \approx 0.001\% \) - **c.)** \( P(21 < x < 35) \approx 22.66\% \)