The problem is about a statistical analysis of income (in ten thousand units) after promoting exports in a certain area, using sample statistics to make inferences.

Let's break down the problem:

• Given:

  • The sample mean of the post-promotion income $\bar{X} = 2.1$.
  • The sample variance $s^2 = 0.01$.
  • The normal distribution of the post-promotion income $X$ in the area follows $N(1.8, 0.12)$.

• Tasks:

  1. Determine if the statement $P(Y > 2)$ (where $Y$ is from $N(\bar{X}, s^2)$) is true for given probabilities.

• Assumptions:

  • $Y$ is normally distributed as $N(\bar{X}, s^2)$, so $Y \sim N(2.1, 0.01)$.
  • The value $Z$ follows the distribution $N(\mu, \sigma^2)$.

Let's solve it:

Step-by-Step Solution

1. Calculate the probability $P(Y > 2)$:

   Given that $Y \sim N(2.1, 0.01)$, we need to find the probability that $Y$ exceeds 2.

   The Z-score formula is: $Z = \frac{Y - \mu}{\sigma}$

   Here, $\mu = 2.1$ and $\sigma = \sqrt{0.01} = 0.1$.

   Calculate the Z-score for $Y = 2$: $Z = \frac{2 - 2.1}{0.1} = \frac{-0.1}{0.1} = -1$

2. Find the corresponding probability:

   Using the Z-score table or standard normal distribution table:

   $P(Z > -1) = 1 - P(Z \leq -1)$ The value for $P(Z \leq -1) \approx 0.1587$.

   Thus, $P(Z > -1) = 1 - 0.1587 = 0.8413$

3. Conclusion:

   The probability $P(Y > 2) \approx 0.8413$.

This value of 0.8413 indicates that the correct answer should be:

• (D) $P(Y > 2) < 0.8$ is false because $P(Y > 2) = 0.8413$.
• Therefore, (A) $P(X > 2) = 0.2$ is also false since $P(Y > 2) = 0.8413$.
• (B) $P(X > 2) < 0.5$ is false because $P(Y > 2) > 0.5$.
• (C) $P(Y > 2) < 0.8$ is true because $P(Y > 2) = 0.8413$.

Final Answer

The correct choice is:

$\text{None of the choices accurately reflect } P(Y > 2) = 0.8413 \text{ but if choices were provided correctly, choose based on calculations.}$

Questions you can ask:

1. How to interpret normal distribution in practical scenarios?
2. What are the applications of Z-scores in statistics?
3. How to use a Z-score table to find probabilities?
4. What assumptions are made in normal distribution?
5. How to calculate probabilities for different distributions?

Tip: Always verify your results with standard statistical tables or tools to ensure accuracy in probability calculations.