1. **State the problem:** We have 25 incubation periods (in hours) from a foodborne disease outbreak: 18, 20, 22, 22, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 35, 36, 38, 40, 42, 45, 72. 2. **Calculate the arithmetic mean:** The mean is the sum of all values divided by the number of values. $$\text{Mean} = \frac{18 + 20 + 22 + 22 + 24 + 24 + 25 + 26 + 26 + 27 + 28 + 28 + 29 + 30 + 30 + 31 + 32 + 33 + 35 + 36 + 38 + 40 + 42 + 45 + 72}{25}$$ Calculate the sum: $$18 + 20 + 22 + 22 + 24 + 24 + 25 + 26 + 26 + 27 + 28 + 28 + 29 + 30 + 30 + 31 + 32 + 33 + 35 + 36 + 38 + 40 + 42 + 45 + 72 = 823$$ So, $$\text{Mean} = \frac{823}{25} = 32.92$$ hours. 3. **Calculate the median:** The median is the middle value when data is ordered. Since there are 25 values (odd number), the median is the 13th value. Ordered data is already given. The 13th value is 29. So, $$\text{Median} = 29$$ hours. 4. **Calculate the mode:** The mode is the most frequent value(s). Frequencies: - 22 appears twice - 24 appears twice - 26 appears twice - 28 appears twice - 30 appears twice All these values appear twice, so the data is multimodal with modes: 22, 24, 26, 28, 30. 5. **Best measure to represent typical incubation period:** The mean is influenced by the outlier 72, which is much larger than other values, so it may not represent the typical case well. The median (29) is less affected by outliers and better represents the typical incubation period. 6. **Calculate the range:** Range = max - min = 72 - 18 = 54 hours. 7. **Calculate the standard deviation:** First, calculate each deviation from the mean, square it, sum all, then divide by $n-1=24$, and take the square root. $$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$$ Calculate squared deviations sum: $$\sum (x_i - 32.92)^2 = (18-32.92)^2 + (20-32.92)^2 + \cdots + (72-32.92)^2$$ Calculations: $(18-32.92)^2 = 219.0864$ $(20-32.92)^2 = 167.3264$ $(22-32.92)^2 = 119.6864$ (twice) total $239.3728$ $(24-32.92)^2 = 79.6864$ (twice) total $159.3728$ $(25-32.92)^2 = 62.7264$ $(26-32.92)^2 = 47.3664$ (twice) total $94.7328$ $(27-32.92)^2 = 35.0464$ $(28-32.92)^2 = 24.2064$ (twice) total $48.4128$ $(29-32.92)^2 = 15.3664$ $(30-32.92)^2 = 8.5264$ (twice) total $17.0528$ $(31-32.92)^2 = 3.6864$ $(32-32.92)^2 = 0.8464$ $(33-32.92)^2 = 0.0064$ $(35-32.92)^2 = 4.3264$ $(36-32.92)^2 = 9.4864$ $(38-32.92)^2 = 25.8064$ $(40-32.92)^2 = 50.0864$ $(42-32.92)^2 = 82.3664$ $(45-32.92)^2 = 146.0864$ $(72-32.92)^2 = 1521.4464$ Sum all: $$219.0864 + 167.3264 + 239.3728 + 159.3728 + 62.7264 + 94.7328 + 35.0464 + 48.4128 + 15.3664 + 17.0528 + 3.6864 + 0.8464 + 0.0064 + 4.3264 + 9.4864 + 25.8064 + 50.0864 + 82.3664 + 146.0864 + 1521.4464 = 2697.08$$ Divide by 24: $$\frac{2697.08}{24} = 112.3783$$ Standard deviation: $$s = \sqrt{112.3783} = 10.6$$ hours (rounded). **Final answers:** - Mean = 32.92 hours - Median = 29 hours - Mode = 22, 24, 26, 28, 30 hours (multimodal) - Range = 54 hours - Standard deviation = 10.6 hours The median best represents the typical incubation period because it is less affected by the outlier 72 hours.