To solve the problem, we need to calculate the probability $P(X > 35)$, where $X$ is a normally distributed variable with a mean $\mu = 50$ and a standard deviation $\sigma = 7$.

Step 1: Standardize the Variable

We convert $X = 35$ into a standard normal variable $Z$ using the formula: $Z = \frac{X - \mu}{\sigma}$ Substituting the values: $Z = \frac{35 - 50}{7} = \frac{-15}{7} \approx -2.14$

Step 2: Find $P(Z > -2.14)$

Using the symmetry of the standard normal distribution: $P(Z > -2.14) = 1 - P(Z \leq -2.14)$

From standard normal tables or a calculator, the cumulative probability $P(Z \leq -2.14)$ is approximately 0.0162.

Thus: $P(Z > -2.14) = 1 - 0.0162 = 0.9838$

Step 3: Interpret the Result

The probability $P(X > 35)$ is approximately 0.9838.

Would you like a detailed breakdown or graph of the normal distribution for better understanding?

Here are five related questions for practice:

1. What is $P(X < 45)$ for the same distribution?
2. How do you calculate $P(40 < X < 60)$ for a normal distribution?
3. What is the probability $P(X > 60)$?
4. How do you interpret $P(X > \mu)$ for any normal distribution?
5. How do you find probabilities for non-standard normal distributions?

Tip: Always standardize the random variable when working with normal distributions to make use of standard normal tables or calculators!