We are given that the random variable $X$ is normally distributed with $\mu = 75$ and $\sigma = 5$, meaning:

$X \sim N(75, 5^2)$

1. Find $P(X > 85)$

We first standardize the value of 85 by converting it to the standard normal form $Z$, where: $Z = \frac{X - \mu}{\sigma}$

So, for $X = 85$: $Z = \frac{85 - 75}{5} = \frac{10}{5} = 2$

Now, we need to find $P(X > 85) = P(Z > 2)$.

Using standard normal tables or a calculator: $P(Z > 2) = 1 - P(Z \leq 2) \approx 1 - 0.9772 = 0.0228$

Thus: $P(X > 85) \approx 0.0228$

2. Find $k$ such that $P(X < k) = 0.997$

We need to find the value of $k$ such that $P(X < k) = 0.997$. This means finding the corresponding $Z$-value for which the cumulative probability is 0.997.

Using standard normal tables, we find that: $P(Z < 2.747) \approx 0.997$

Now, convert the $Z$-value back to $X$ using: $k = Z \cdot \sigma + \mu$

So: $k = 2.747 \cdot 5 + 75 = 13.735 + 75 = 88.735$

Thus, the value of $k$ is approximately 88.74. This means that 99.7% of the distribution of $X$ lies below 88.74.

3. Find the two values where the middle 68.27% of the distribution lies

The middle 68.27% of the distribution corresponds to one standard deviation above and below the mean in a normal distribution.

For the standard normal distribution, this means: $\mu - \sigma \quad \text{to} \quad \mu + \sigma$

Thus: $75 - 5 = 70 \quad \text{and} \quad 75 + 5 = 80$

So, the middle 68.27% of the distribution lies between 70 and 80.

Summary of Results:

1. $P(X > 85) \approx 0.0228$
2. $k \approx 88.74$ such that $P(X < k) = 0.997$
3. The middle 68.27% of the distribution lies between 70 and 80.

Would you like more details or clarification on any part of this?

Here are some related questions to expand on this topic:

1. How do you compute probabilities for other values of $X$?
2. What is the 95th percentile for this distribution?
3. How do changes in $\mu$ and $\sigma$ affect the distribution?
4. How can you interpret the result for $P(X > 85)$ in practical terms?
5. What is the probability that $X$ is between 80 and 85?

Tip: When dealing with normal distributions, always standardize using the formula $Z = \frac{X - \mu}{\sigma}$ to simplify the calculations!