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(34<X<63) Which of the following normal curves corresponds to P(34<X<63)?

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 \( P(34 < X < 63) \).

To find the probability, we first convert the values 34 and 63 to their corresponding z-scores using the formula: \[ z = \frac{X - \mu}{\sigma} \]

For \( X = 34 \): \[ z_1 = \frac{34 - 50}{7} = \frac{-16}{7} \approx -2.29 \]

For \( X = 63 \): \[ z_2 = \frac{63 - 50}{7} = \frac{13}{7} \approx 1.86 \]

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

For \( z_1 \approx -2.29 \): \[ P(Z < -2.29) \approx 0.0110 \]

For \( z_2 \approx 1.86 \): \[ P(Z < 1.86) \approx 0.9686 \]

The probability \( P(34 < X < 63) \) is the difference between the two probabilities: \[ P(34 < X < 63) = P(Z < 1.86) - P(Z < -2.29) \] \[ P(34 < X < 63) = 0.9686 - 0.0110 = 0.9576 \]

The correct normal curve corresponds to the area between \( X = 34 \) and \( X = 63 \). This is represented by the shaded area under the curve between these two points.

Final Answer