According to the sampling distribution of the mean:
The sample mean $\bar{X}$ is an unbiased, consistent, and minimum-variance estimator of the population mean $\mu$.
When repeated samples are drawn, the average of multiple sample means (the grand mean $\bar{\bar{X}}$) pools information across all samples, drastically reducing variance ($\text{Var}(\bar{\bar{X}}) = \frac{\sigma^2}{k \cdot n}$) and converging directly to $\mu$.
A single observation (Option 2) or small sample median (Option 1) has much higher variance.

Analysis of regression properties under transformation:

• Incorrect: Adding a constant does not increase the slope.
• Incorrect: Division by 35 occurs only if variables are scaled by multiplying by 35.
• Incorrect: Subtraction does not affect the slope.
• Correct: Regression coefficients ($b_{yx} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}$) are strictly independent of the change of origin. Adding any constant (here 35) to all values of $X$ and $Y$ shifts the mean but leaves deviations $(X - \bar{X})$ and $(Y - \bar{Y})$ completely unchanged. Hence, the regression coefficient remains exactly the same as the original.

Analysis of linear line slopes:
In the equation $Y = a + bX$, $b$ represents the slope $\frac{dY}{dX}$.

• Incorrect: $b \ne 0$ produces an inclined line.
• Incorrect: $b = 1$ gives an angle of $45^\circ$ with the horizontal.
• Correct: A line is completely horizontal (parallel to the X-axis) when its slope is zero ($b = 0$). In this case, $Y = a$, meaning $Y$ remains constant regardless of $X$.
• Incorrect: $a = b$ specifies a relationship between the intercept and slope, not horizontality.

Analysis of properties of normal distribution:

• Correct (Not a characteristic): In a general normal distribution $N(\mu, \sigma^2)$, the mean $\mu$ can be any real number ($-\infty \lt \mu \lt \infty$). It is zero only in the special case of the standard normal distribution $N(0, 1)$.
• Incorrect (Is a characteristic): The total area under the probability density curve equals exactly 1 (total probability).
• Incorrect (Is a characteristic): Because the distribution is bell-shaped and unimodal symmetric, $\text{Mean} = \text{Median} = \text{Mode} = \mu$.
• Incorrect (Is a characteristic): The normal distribution curve is perfectly symmetrical about the vertical ordinate at $x = \mu$.