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. 1 of P(X>38) Click the icon to view a table of areas under the normal curve. Which of the following normal curves corresponds to P(X>38) ? A. B. c. P(X>38)= (Round to four decimal places as needed.)

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.

1 of

P(X>38)

Click the icon to view a table of areas under the normal curve.

Which of the following normal curves corresponds to P(X>38) ?
A.
B.
c.

P(X>38)=

(Round to four decimal places as needed.)

Transcript text: Assume the random variable $X$ is normally distributed with mean $\mu=50$ and standard deviation $\sigma=7$. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded. 1 of \[ P(X>38) \] Click the icon to view a table of areas under the normal curve. Which of the following normal curves corresponds to $P(X>38)$ ? A. B. c. \[ P(X>38)= \] $\square$ (Round to four decimal places as needed.)

Solution

Solution Steps

Step 1: Identify the given parameters

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(X > 38) $.

Step 2: Standardize the variable

To find $ P(X > 38) $, we first convert the value 38 to a z-score using the formula: \[ z = \frac{X - \mu}{\sigma} \] Substituting the given values: \[ z = \frac{38 - 50}{7} = \frac{-12}{7} \approx -1.7143 \]

Step 3: Use the z-score to find the probability

We need to find the probability that $ Z $ (the standard normal variable) is greater than -1.7143. Using the standard normal distribution table or a calculator, we find the cumulative probability up to $ z = -1.7143 $.

The cumulative probability for $ z = -1.7143 $ is approximately 0.0432. Since we want $ P(Z > -1.7143) $: \[ P(Z > -1.7143) = 1 - P(Z \leq -1.7143) = 1 - 0.0432 = 0.9568 \]

Step 4: Identify the correct normal curve

The correct normal curve corresponding to $ P(X > 38) $ is the one where the area to the right of 38 is shaded. This corresponds to option C.