1. **Problem Statement:** We are given two datasets and need to calculate the mean and sample standard deviation for each, then analyze the effect of an outlier (100) on these statistics and identify which statistics are resistant to outliers. 2. **Mean Formula:** The mean of a dataset with $n$ values $x_1, x_2, ..., x_n$ is given by: $$\text{mean} = \frac{1}{n} \sum_{i=1}^n x_i$$ 3. **Calculate Mean for Dataset I:** Dataset I: 1, 1, 5, 12, 14, 16, 18, 19, 19 Number of values $n=9$ $$\text{mean}_I = \frac{1+1+5+12+14+16+18+19+19}{9} = \frac{105}{9} = 11.67$$ 4. **Calculate Mean for Dataset II:** Dataset II: 1, 1, 5, 12, 14, 16, 18, 19, 100 Number of values $n=9$ $$\text{mean}_{II} = \frac{1+1+5+12+14+16+18+19+100}{9} = \frac{186}{9} = 20.67$$ 5. **Effect of Outlier on Mean:** The mean increased from 11.67 to 20.67 due to the outlier 100, which is a significant change. 6. **Sample Standard Deviation Formula:** The sample standard deviation $s$ is: $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}$$ where $\bar{x}$ is the sample mean. 7. **Calculate Sample Standard Deviation for Dataset I:** Calculate squared deviations: $(1-11.67)^2=113.44$, $(1-11.67)^2=113.44$, $(5-11.67)^2=44.49$, $(12-11.67)^2=0.11$, $(14-11.67)^2=5.44$, $(16-11.67)^2=18.77$, $(18-11.67)^2=40.11$, $(19-11.67)^2=53.44$, $(19-11.67)^2=53.44$ Sum of squared deviations: $$113.44+113.44+44.49+0.11+5.44+18.77+40.11+53.44+53.44=442.68$$ Divide by $n-1=8$: $$\frac{442.68}{8} = 55.34$$ Standard deviation: $$s_I = \sqrt{55.34} = 7.44$$ 8. **Calculate Sample Standard Deviation for Dataset II:** Squared deviations: $(1-20.67)^2=384.11$, $(1-20.67)^2=384.11$, $(5-20.67)^2=244.49$, $(12-20.67)^2=75.11$, $(14-20.67)^2=44.49$, $(16-20.67)^2=21.78$, $(18-20.67)^2=7.11$, $(19-20.67)^2=2.78$, $(100-20.67)^2=6,239.11$ Sum: $$384.11+384.11+244.49+75.11+44.49+21.78+7.11+2.78+6239.11=7,403.09$$ Divide by $8$: $$\frac{7,403.09}{8} = 925.39$$ Standard deviation: $$s_{II} = \sqrt{925.39} = 30.42$$ 9. **Effect of Outlier on Standard Deviation:** The standard deviation increased from 7.44 to 30.42, showing a significant effect of the outlier. 10. **Statistics Resistant to Outliers:** - The mean and standard deviation are sensitive to outliers. - The median and interquartile range (IQR) are resistant to outliers. **Final answers:** (g) Mean of Dataset I: 11.67 (h) Mean of Dataset II: 20.67 (i) Does the outlier significantly affect the mean? Yes (j) Sample standard deviation of Dataset I: 7.44 (k) Sample standard deviation of Dataset II: 30.42 (l) Does the outlier significantly affect the standard deviation? Yes (m) Statistics resistant to outliers: IQR, median