1. **Problem Statement:** We analyze sales data of 8 outlets to compute descriptive statistics, construct a frequency distribution, and calculate probabilities assuming normal distribution. 2. **Given Data:** Sales (MK Million): $45, 52, 39, 60, 55, 48, 62, 50$ ### a) Compute mean, median, mode, range, variance, and standard deviation 3. **Mean formula:** $$\text{Mean} = \frac{\sum x_i}{n}$$ Calculate sum: $45 + 52 + 39 + 60 + 55 + 48 + 62 + 50 = 411$ Number of outlets $n=8$ $$\text{Mean} = \frac{411}{8} = 51.375$$ 4. **Median:** Sort data: $39, 45, 48, 50, 52, 55, 60, 62$ Median for even $n$ is average of middle two: $$\frac{50 + 52}{2} = 51$$ 5. **Mode:** No repeated values, so no mode. 6. **Range:** $$\text{Range} = \text{max} - \text{min} = 62 - 39 = 23$$ 7. **Variance formula:** $$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$$ Calculate squared deviations: $(45-51.375)^2=40.64$, $(52-51.375)^2=0.39$, $(39-51.375)^2=153.14$, $(60-51.375)^2=74.39$, $(55-51.375)^2=13.14$, $(48-51.375)^2=11.39$, $(62-51.375)^2=112.89$, $(50-51.375)^2=1.89$ Sum: $40.64+0.39+153.14+74.39+13.14+11.39+112.89+1.89=407.87$ $$s^2 = \frac{407.87}{7} = 58.27$$ 8. **Standard deviation:** $$s = \sqrt{58.27} = 7.63$$ ### b) Frequency distribution table (class interval = 5) 9. Classes: 35-39, 40-44, 45-49, 50-54, 55-59, 60-64 Count frequencies: 35-39: 1 (39) 40-44: 0 45-49: 3 (45,48,50) 50-54: 1 (52) 55-59: 1 (55) 60-64: 2 (60,62) | Class Interval | Frequency | |----------------|-----------| | 35 - 39 | 1 | | 40 - 44 | 0 | | 45 - 49 | 3 | | 50 - 54 | 1 | | 55 - 59 | 1 | | 60 - 64 | 2 | Histogram can be drawn using these frequencies. ### c) Normal distribution probabilities Assuming $\mu=51.375$, $\sigma=7.63$ 10. **(i) Probability sales > 55** Calculate z-score: $$z = \frac{55 - 51.375}{7.63} = 0.48$$ From standard normal table, $P(Z > 0.48) = 1 - 0.6844 = 0.3156$ 11. **(ii) Probability sales between 45 and 55** Calculate z-scores: $$z_1 = \frac{45 - 51.375}{7.63} = -0.84$$ $$z_2 = \frac{55 - 51.375}{7.63} = 0.48$$ From table: $P(Z < 0.48) = 0.6844$ $P(Z < -0.84) = 0.2005$ Probability between: $$0.6844 - 0.2005 = 0.4839$$ **Final answers:** - Mean = 51.375 - Median = 51 - Mode = None - Range = 23 - Variance = 58.27 - Standard deviation = 7.63 - Frequency table as above - $P(X > 55) = 0.3156$ - $P(45 < X < 55) = 0.4839$