sume the random variable \( x \) is normally distributed with mean \( \mu=50 \) and standard deviation \( \sigma=7 \). Find the indicate obability. \( P(x>37) \) \( (x>37)=\square \) ound to four decimal places as needed.)

To find the probability \( P(x > 37) \) for a normally distributed random variable \( x \) with mean \( \mu = 50 \) and standard deviation \( \sigma = 7 \), follow these steps: 1. **Standardize the Variable:** Convert the original variable \( x \) to a standard normal variable \( z \) using the formula: \[ z = \frac{x - \mu}{\sigma} \] For \( x = 37 \): \[ z = \frac{37 - 50}{7} = \frac{-13}{7} \approx -1.8571 \] 2. **Find the Probability:** We need to find \( P(x > 37) \), which is equivalent to \( P(z > -1.8571) \). Due to the symmetry of the normal distribution: \[ P(z > -1.8571) = 1 - P(z < -1.8571) = P(z < 1.8571) \] Using standard normal distribution tables or a calculator, find \( \Phi(1.8571) \), where \( \Phi \) denotes the cumulative distribution function (CDF) of the standard normal distribution. 3. **Lookup or Calculate \( \Phi(1.8571) \):** - For \( z = 1.85 \), \( \Phi(1.85) \approx 0.9678 \) - For \( z = 1.86 \), \( \Phi(1.86) \approx 0.9686 \) Interpolating for \( z = 1.8571 \): \[ \Phi(1.8571) \approx 0.9678 + \frac{0.0008}{0.01} \times 0.0071 \approx 0.9684 \] 4. **Conclusion:** \[ P(x > 37) = 0.9684 \] **Final Answer:** \[ P(x > 37) = 0.9684 \]