Questions: Assume the random variable X is normally distributed with mean μ=50 and standard deviation σ=7. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded. P(35<X<66) Which of the following normal curves corresponds to P(35<X<66) ? A. B. C. P(35<X<66)= (Round to four decimal places as needed.)

Solution Steps

The problem states that the random variable \( X \) is normally distributed with a mean \( \mu = 50 \) and a standard deviation \( \sigma = 7 \). We need to compute the probability \( P(35 < X < 66) \).

To find the probability, we first convert the values 35 and 66 to their corresponding z-scores using the formula: \[ z = \frac{X - \mu}{\sigma} \]

For \( X = 35 \): \[ z_1 = \frac{35 - 50}{7} = \frac{-15}{7} \approx -2.14 \]

For \( X = 66 \): \[ z_2 = \frac{66 - 50}{7} = \frac{16}{7} \approx 2.29 \]

Next, we use the standard normal distribution table to find the probabilities corresponding to the z-scores.

For \( z_1 = -2.14 \): \[ P(Z < -2.14) \approx 0.0162 \]

For \( z_2 = 2.29 \): \[ P(Z < 2.29) \approx 0.9890 \]

The probability \( P(35 < X < 66) \) is the difference between the probabilities found in the standard normal distribution table: \[ P(35 < X < 66) = P(Z < 2.29) - P(Z < -2.14) \] \[ P(35 < X < 66) = 0.9890 - 0.0162 = 0.9728 \]

Final Answer