**The Number of Dogs per Household in a Small Town** | Dogs | 0 | 1 | 2 | 3 | 4 | 5 | |--------------|-------|-------|-------|-------|-------|-------| | Probability | 0.632 | 0.228 | 0.092 | 0.027 | 0.014 | 0.008 | - **σ = ___** (Round to one decimal place as needed.) **(b) Interpret the results in the context of the real-life situation.** - **Options:** - **A.** A household on average has 0.9 dog with a standard deviation of 0.6 dog. - **B.** A household on average has 0.6 dog with a standard deviation of 14 dogs. - **C.** A household on average has 0.9 dog with a standard deviation of 0.9 dog. - **D.** A household on average has 0.6 dog with a standard deviation of 1.0 dog. **Explanation of Data Representation:** - **Table Overview:** - The table represents the probability distribution of the number of dogs per household in a small town. - For each number of dogs (from 0 to 5), the corresponding probability is listed, indicating how likely it is for a household to have that many dogs. - **Interpretation of Options:** - Option A suggests an average (mean) number of dogs per household together with its variability, indicated by the standard deviation. - Each option provides different interpretations based on statistical mean and standard deviation calculations. **Title: Analyzing the Probability Distribution of Dogs Per Household** **Objective:** To calculate the mean, variance, and standard deviation of a probability distribution involving the number of dogs per household in a small town. **Probability Distribution Table:** | Number of Dogs (X) | Probability (P(X)) | |--------------------|--------------------| | 0 | 0.632 | | 1 | 0.228 | | 2 | 0.092 | | 3 | 0.027 | | 4 | 0.014 | | 5 | 0.008 | **Tasks:** (a) **Find the Mean of the Probability Distribution:** - The mean (μ) is the expected value of the distribution, calculated by summing the products of each value and its corresponding probability. \[ \mu = \sum (X \cdot P(X)) \] (b) **Calculate the Variance of the Probability Distribution:** - The variance (σ²) measures the spread of the values in the distribution, calculated using: \[ \sigma^2 = \sum ((X - \mu)^2 \cdot P(X)) \] (c) **Determine the Standard Deviation:** - The standard deviation (σ) is the square root of the variance, providing insight into the data's dispersion. \[ \sigma = \sqrt{\sigma^2} \] **Instructions:** - Perform each calculation and round answers to one decimal place as needed. - The mean provides insight into the average number of dogs per household. - The variance and standard deviation help in understanding the variability of the data. **Note:** Understanding the calculation of these statistical measures is crucial for interpreting data distributions effectively. Use a calculator for more accurate results.