1. **Problem Statement:** Calculate the mean, median, mode, range, variance, and standard deviation for the time (in minutes) it takes smokers to fall asleep given the data: 69.3, 56.0, 22.1, 47.6, 53.2, 48.1, 52.7, 60.2, 43.8, 34.4. 2. **Mean Formula:** The mean is the sum of all data points divided by the number of points. $$\text{Mean} = \frac{\sum x_i}{n}$$ 3. **Calculate Mean:** $$\sum x_i = 69.3 + 56.0 + 22.1 + 47.6 + 53.2 + 48.1 + 52.7 + 60.2 + 43.8 + 34.4 = 487.4$$ $$n = 10$$ $$\text{Mean} = \frac{487.4}{10} = 48.74$$ 4. **Median Formula:** The median is the middle value when data is ordered. For even $n$, median is average of two middle values. 5. **Calculate Median:** Order data: 22.1, 34.4, 43.8, 47.6, 48.1, 52.7, 53.2, 56.0, 60.2, 69.3 Middle values (5th and 6th): 48.1 and 52.7 $$\text{Median} = \frac{48.1 + 52.7}{2} = 50.4$$ 6. **Mode:** No repeated values, so no mode. 7. **Range Formula:** $$\text{Range} = \text{max} - \text{min}$$ 8. **Calculate Range:** $$\text{Range} = 69.3 - 22.1 = 47.2$$ 9. **Variance Formula:** $$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$$ 10. **Calculate Variance:** Calculate each squared deviation: $(69.3 - 48.74)^2 = 420.67$ $(56.0 - 48.74)^2 = 52.67$ $(22.1 - 48.74)^2 = 705.38$ $(47.6 - 48.74)^2 = 1.30$ $(53.2 - 48.74)^2 = 20.83$ $(48.1 - 48.74)^2 = 0.41$ $(52.7 - 48.74)^2 = 15.68$ $(60.2 - 48.74)^2 = 131.14$ $(43.8 - 48.74)^2 = 24.40$ $(34.4 - 48.74)^2 = 205.15$ Sum of squared deviations: $$\sum (x_i - \bar{x})^2 = 1577.63$$ $$s^2 = \frac{1577.63}{9} = 175.29$$ 11. **Standard Deviation:** $$s = \sqrt{175.29} = 13.24$$ **Final answers:** - Mean = 48.74 - Median = 50.4 - Mode = None - Range = 47.2 - Variance = 175.29 - Standard Deviation = 13.24