Let's break down and solve each part of the problem step by step.

Given Information:

• $\mu = 75$ (mean)
• $\sigma = 5$ (standard deviation)
• The random variable $X$ follows a normal distribution, i.e., $X \sim N(\mu = 75, \sigma = 5)$.

1. Find $P(X > 85)$:

We are looking for the probability that $X$ is greater than 85.

First, we convert this to a standard normal variable $Z$ using the formula: $Z = \frac{X - \mu}{\sigma}$ For $X = 85$: $Z = \frac{85 - 75}{5} = 2$

Now, $P(X > 85) = P(Z > 2)$.

Using standard normal distribution tables (or a calculator), we know that: $P(Z > 2) \approx 0.0228$

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

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

This means we need to find the value of $k$ where 99.7% of the distribution lies to the left of $k$. This corresponds to finding the 99.7th percentile of the normal distribution.

From the standard normal distribution tables or a calculator, the Z-value corresponding to a cumulative probability of 0.997 is approximately 2.747.

Now, convert this Z-value back to the original $X$ distribution: $k = \mu + Z \cdot \sigma = 75 + 2.747 \cdot 5 \approx 75 + 13.735 = 88.735$

Thus, $k \approx 88.74$.

Interpretation: There is a 99.7% probability that the value of $X$ is less than or equal to approximately 88.74.

3. Find the two values where the middle 68.27% of the distribution of $X$ lie:

The middle 68.27% of the distribution corresponds to 1 standard deviation on either side of the mean in a normal distribution (since 68.27% is the empirical rule for $1 \sigma$).

So, we calculate the range of values that are within 1 standard deviation from the mean: $\mu - \sigma = 75 - 5 = 70$ $\mu + \sigma = 75 + 5 = 80$

Therefore, the two values are 70 and 80.

Interpretation: 68.27% of the values of $X$ will lie between 70 and 80.

Summary of Results:

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

Would you like any further details or have questions on these calculations?

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Here are 5 relative questions to expand the understanding:

1. How do we calculate probabilities for values other than 85 in a normal distribution?
2. How does changing $\mu$ or $\sigma$ affect the distribution and probabilities?
3. What is the interpretation of the 95% confidence interval in a normal distribution?
4. How would the probability change if we wanted $P(X < 65)$?
5. What is the probability that $X$ lies between 70 and 80?

Tip: In normal distributions, about 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations from the mean, respectively. This is known as the empirical rule.