1. **Distinguish between qualitative and quantitative variables:** Qualitative variables describe categories or qualities (e.g., color, gender) and are non-numeric. Quantitative variables represent numeric values and can be measured or counted (e.g., height, weight). 2. **Write the following using summation sign with appropriate index:** a) $Y_3 + Y_4 + \cdots + Y_{15} = \sum_{i=3}^{15} Y_i$ b) $Y_1^2 + Y_2^2 + Y_3^2 + Y_4^2 = \sum_{i=1}^4 Y_i^2$ c) $(Y_1 - \mu)^2 + (Y_2 - \mu)^2 + (Y_3 - \mu)^2 = \sum_{i=1}^3 (Y_i - \mu)^2$ d) $bY_{20} + bY_{21} + \cdots + bY_{30} = b \sum_{i=20}^{30} Y_i$ 3. **Expand the following summation and product signs:** a) $\sum_{i=1}^7 Y_i = Y_1 + Y_2 + Y_3 + Y_4 + Y_5 + Y_6 + Y_7$ b) $\sum_{i=5}^8 (Y_i - \mu) = (Y_5 - \mu) + (Y_6 - \mu) + (Y_7 - \mu) + (Y_8 - \mu)$ c) $\sum_{i=1}^7 Y_i^2 = Y_1^2 + Y_2^2 + Y_3^2 + Y_4^2 + Y_5^2 + Y_6^2 + Y_7^2$ d) $\sum_{i=2}^6 (Y_i - 9) = (Y_2 - 9) + (Y_3 - 9) + (Y_4 - 9) + (Y_5 - 9) + (Y_6 - 9)$ e) $\prod_{i=1}^n (X_i Y_i) = X_1 Y_1 \times X_2 Y_2 \times \cdots \times X_n Y_n$ f) $\prod_{i=3}^7 (a Y_i)^2 = (a Y_3)^2 \times (a Y_4)^2 \times (a Y_5)^2 \times (a Y_6)^2 \times (a Y_7)^2$ 4. **Classify the following variables:** i) Sex of an insect: Categorical (qualitative) ii) Weights of plants: Continuous (quantitative) iii) Major crops of Pakistan: Categorical iv) Level of satisfaction: Ordinal categorical (qualitative) v) Teaching standards: Ordinal categorical vi) Temperature measured in Fahrenheit: Continuous 5. **Explain main aspects of a Statistical problem:** - **Data Collection:** Gathering relevant data accurately. - **Data Description:** Summarizing data using tables, graphs, and statistics. - **Data Analysis:** Applying statistical methods to interpret data. - **Inference:** Drawing conclusions or making predictions based on data. - **Decision Making:** Using results to inform actions or policies. Each step is crucial to ensure valid, reliable, and meaningful statistical conclusions.