To find the probability $P(x > 36)$ for a normally distributed random variable $x$ with mean $\mu = 50$ and standard deviation $\sigma = 7$, follow these steps:

Step 1: Standardize the value

We first convert $x = 36$ to a z-score using the formula: $z = \frac{x - \mu}{\sigma}$ Substitute the values $\mu = 50$ and $\sigma = 7$: $z = \frac{36 - 50}{7} = \frac{-14}{7} = -2$

Step 2: Use the z-score to find the probability

Now, we need to find $P(x > 36) = P(z > -2)$. Using standard normal distribution tables or a calculator, we can look up the cumulative probability for $z = -2$.

The cumulative probability for $z = -2$ is approximately 0.0228. This represents $P(z \leq -2)$.

Step 3: Find the desired probability

Since we're looking for $P(z > -2)$, we subtract this value from 1: $P(z > -2) = 1 - P(z \leq -2) = 1 - 0.0228 = 0.9772$

Final Answer:

$P(x > 36) = 0.9772$

Would you like more details or have any questions?

Here are five related questions for practice:

1. What is $P(x < 36)$ for the same distribution?
2. Find $P(40 < x < 60)$ given $\mu = 50$ and $\sigma = 7$.
3. How do you calculate $P(x > 64)$?
4. What is the z-score for $x = 55$?
5. For $P(x < 45)$, what value do you get?

Tip: When using z-scores, remember that the standard normal distribution is symmetrical around 0, so probabilities for negative and positive z-scores mirror each other.