Statistical Concepts

Step 1: Understanding standard error.
The standard error (SE) is a measure of the precision of an estimate. It quantifies the amount of variation or dispersion of a sample statistic (such as the sample mean) from the population parameter.

Step 2: Analysis of options.
- (A) Specification error of the model: This is incorrect. Specification error refers to mistakes in the model's form, not the precision of the estimate.
- (B) Autocorrelation in the regression model: This is incorrect. Autocorrelation refers to the correlation of residuals across time, not the precision of the estimate.
- (C) Correlation between dependent and independent variables: This is incorrect. Standard error measures the precision, not the correlation.
- (D) Precision of an estimate: This is correct. The standard error quantifies how precise an estimate is, indicating the variability in the sample mean from the population mean.

Step 3: Conclusion.
The correct answer is (D), as standard error measures the precision of an estimate.

Step 1: Calculate the mean.
The mean ( ) of the sample is:

Step 2: Calculate the squared differences.
- For 8: - For 7: - For 9:

Step 3: Calculate the variance.
Variance ( ) is the average of the squared differences:

Step 4: Calculate the standard deviation.
The standard deviation ( ) is the square root of the variance:

Step 5: Conclusion.
The correct standard deviation is , so the answer is (A).

Step 1: Understanding the relationship between AM, Median, and Mode.
- **Positively Skewed (A):** In a positively skewed distribution, the tail is stretched to the right. In this case, the mode is less than the median, and the median is less than the mean. Therefore, the correct relationship is **AM > Median > Mode**. This corresponds to **(III)** in List-II.
- **Moderately Skewed (B):** A moderately skewed distribution has a small skew. The relationship between the mean, median, and mode is typically **AM < Median < Mode**. This corresponds to **(IV)** in List-II.
- **Negatively Skewed (C):** In a negatively skewed distribution, the tail is stretched to the left. The relationship is given by **AM - Mode = 3 (AM - Median)**. This corresponds to **(II)** in List-II.
- **Symmetric Distribution (D):** In a symmetric distribution, the mean, median, and mode are all equal. This corresponds to **(I)** in List-II.

Step 2: Conclusion.
The correct answer is **(A) - (III), (B) - (IV), (C) - (II), (D) - (I)**.

To calculate the standard deviation, we follow these steps: 1. Find the mean (average): The given sample is 8, 7, 9. The mean is calculated as: Mean = (8 + 7 + 9)/(3) = (24)/(3) = 8 2. Find the squared differences from the mean: Now, subtract the mean from each data point and square the result: (8 - 8)² = 0² = 0 (7 - 8)² = (-1)² = 1 (9 - 8)² = (1)² = 1 3. Calculate the variance: The variance is the average of the squared differences: Variance = (0 + 1 + 1)/(3) = (2)/(3) ≈ 0.6667 4. Find the standard deviation: The standard deviation is the square root of the variance: Standard deviation = √((2)/(3)) ≈ √(2.15) The standard deviation for the sample is approximately √(2.15), which matches option (B). Final Answer: (B) √(2.15).