Assume the random variable X is normally distribut

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<61) Which of the following normal curves corresponds to P(35<X<61) ? A. B. ○ C P(35<X<6=0.9256 (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. \[ P(35

Solution

Solution Steps

Step 1: Find the z-scores

We are given that X follows a normal distribution with mean μ = 50 and standard deviation σ = 7. We want to find P(35 < X < 61). First, we convert the X values to z-scores using the formula: z = (X - μ) / σ.

For X = 35, z = (35 - 50) / 7 = -15/7 ≈ -2.14

For X = 61, z = (61 - 50) / 7 = 11/7 ≈ 1.57

Step 2: Relate the z-scores to the normal curve

The probability P(35 < X < 61) is equivalent to P(-2.14 < Z < 1.57), where Z is a standard normal random variable. This probability corresponds to the area under the standard normal curve between z = -2.14 and z = 1.57. The mean of the standard normal curve is at z = 0, which corresponds to X = 50 on the given graphs.

Step 3: Choose the correct graph

The correct graph will be the one where the shaded area is between X = 35 and X = 61, which corresponds to z-scores approximately -2.14 and 1.57. This shaded region should be centered around the mean, 50.

Option B is the correct graph because the shaded region is between 35 and 61, and is centered on 50.

Step 4: Calculate the probability

P(35 < X < 61) = P(-2.14 < Z < 1.57). This probability is given as 0.9256.