Questions: X is a normally distributed random variable with mean 23 and standard deviation 12. What is the probability that X is between 36 and 68? Write your answer as a decimal rounded to the nearest thousandth.

X is a normally distributed random variable with mean 23 and standard deviation 12. What is the probability that X is between 36 and 68?

Write your answer as a decimal rounded to the nearest thousandth.

Transcript text: $X$ is a normally distributed random variable with mean 23 and standard deviation 12 . What is the probability that $X$ is between 36 and $68 ?$ Write your answer as a decimal rounded to the nearest thousandth. $\square$

Solution

Solution Steps

Step 1: Calculate the CDF at $ x_1 = 36 $

To find the probability that the random variable $ X $ is less than or equal to $ 36 $, we compute the cumulative distribution function (CDF) at this point:

\[ P(X \leq 36) = CDF(36) \approx 0.8607 \]

Step 2: Calculate the CDF at $ x_2 = 68 $

Next, we calculate the CDF at $ x_2 = 68 $:

\[ P(X \leq 68) = CDF(68) \approx 0.9999 \]

Step 3: Find the Probability Between $ x_1 $ and $ x_2 $

The probability that $ X $ is between $ 36 $ and $ 68 $ is given by the difference of the two CDF values:

\[ P(36 < X < 68) = P(X \leq 68) - P(X \leq 36) \approx 0.9999 - 0.8607 = 0.1392 \]