To analyze this problem, let's break it down step-by-step. The given data and the question are regarding the normal distribution and probabilities. Here's a translation and interpretation of the problem and the options:

Given Information:

1. Sample Mean ( $\bar{x}$ ): 2.1
2. Sample Variance ( $s^2$ ): 0.01
3. Population Mean ( $\mu$ ): 1.8
4. Population Variance ( $\sigma^2$ ): 0.12
5. Normal Distribution of Population $Y$ : $N(\bar{x}, s^2) = N(2.1, 0.01)$

The problem is asking about the probabilities related to these normal distributions.

The Question:

Given the conditions and the normal distribution of the output revenue $Y$, we need to evaluate the statements about $P(X > 2)$ and $P(Y > 2)$.

The additional probability given is: $P(Z < \mu + \sigma) = 0.8413$

This implies: $P(Z < \mu + \sigma) = 0.8413$

We know that 0.8413 corresponds to a Z-score of 1 in a standard normal distribution. This implies: $\mu + \sigma = 1$

Calculations:

1. Standard Deviation of the Sample Mean: $s = \sqrt{0.01} = 0.1$

2. Z-score for $X = 2$ using $N(1.8, 0.12)$: $Z = \frac{X - \mu}{\sigma} = \frac{2 - 1.8}{\sqrt{0.12}} \approx \frac{0.2}{0.346} \approx 0.578$

The probability for this Z-score (0.578) can be looked up in Z-tables or calculated using a normal distribution calculator.

3. Z-score for $Y = 2$ using $N(2.1, 0.01)$: $Z = \frac{Y - \bar{x}}{s} = \frac{2 - 2.1}{0.1} = -1$

The probability $P(Y > 2)$ for this Z-score (-1) can be calculated using Z-tables.

Conclusion and Selection:

The question is asking to choose the correct statement based on these probabilities. To do this accurately, let's calculate the approximate probabilities:

1. For $X$ with $N(1.8, 0.12)$: $P(X > 2) \approx 1 - 0.719 = 0.281$

So, $P(X > 2) > 0.2$.

2. For $Y$ with $N(2.1, 0.01)$: $P(Y > 2) \approx 0.8413$

So, $P(Y > 2) < 0.8$ is not correct. But $P(Y > 2) > 0.5$.

So, the correct answers should be:

• $P(X > 2) > 0.2$
• $P(Y > 2) > 0.5$

Thus, the correct options are A and C.

Final Answer:

• A: $P(X > 2) > 0.2$
• C: $P(Y > 2) > 0.5$

Would you like further details on the calculations or any specific part of this solution?

Here are 5 related questions you might find interesting:

1. How do you calculate the Z-score for a given value in a normal distribution?
2. What is the significance of the standard deviation in a normal distribution?
3. How can you use the standard normal distribution table to find probabilities?
4. What are the properties of a normal distribution?
5. How do you interpret the results from a normal distribution in real-world scenarios?

Tip: Always remember to convert the variance to standard deviation by taking its square root before calculating Z-scores.