1. Problem: Classify variables as categorical or numerical, and if numerical, as discrete or continuous, and identify the measurement scale. 1.a Number of telephones per household: - This is a numerical variable because it counts telephones. - It is discrete since telephones are countable in whole numbers. - Measurement scale: Ratio scale (has a true zero and meaningful ratios). 1.b Length (in minutes) of the longest telephone call made in a month: - Numerical variable because it measures time. - Continuous since time can be any value within a range. - Measurement scale: Ratio scale. 1.c Whether someone in the household owns a Wi-Fi-capable cell phone: - Categorical variable (yes/no). - Measurement scale: Nominal scale (categories without order). 1.d Whether there is a high-speed Internet connection in the household: - Categorical variable (yes/no). - Measurement scale: Nominal scale. 2. Problem: Given data set $\{7,4,9,7,3,12\}$ with $n=6$, compute descriptive statistics and analyze data. 2.a Compute mean, median, and mode: - Mean: $$\frac{7+4+9+7+3+12}{6} = \frac{42}{6} = 7$$ - Median: Sort data: $\{3,4,7,7,9,12\}$; median is average of middle two: $$\frac{7+7}{2} = 7$$ - Mode: Most frequent value is 7. 2.b Compute range, variance, standard deviation, coefficient of variation: - Range: $12 - 3 = 9$ - Variance: Calculate squared deviations from mean 7: $$(7-7)^2=0, (4-7)^2=9, (9-7)^2=4, (7-7)^2=0, (3-7)^2=16, (12-7)^2=25$$ - Sum of squared deviations: $0+9+4+0+16+25=54$ - Sample variance: $$s^2 = \frac{54}{6-1} = \frac{54}{5} = 10.8$$ - Standard deviation: $$s = \sqrt{10.8} \approx 3.29$$ - Coefficient of variation: $$\frac{s}{\text{mean}} = \frac{3.29}{7} \approx 0.47$$ 2.c Compute Z scores and check for outliers: - Z score formula: $$Z = \frac{x - \text{mean}}{s}$$ - Calculate for each: - $Z_7 = \frac{7-7}{3.29} = 0$ - $Z_4 = \frac{4-7}{3.29} \approx -0.91$ - $Z_9 = \frac{9-7}{3.29} \approx 0.61$ - $Z_7 = 0$ - $Z_3 = \frac{3-7}{3.29} \approx -1.22$ - $Z_{12} = \frac{12-7}{3.29} \approx 1.52$ - Common outlier rule: $|Z| > 2$; no values exceed this, so no outliers. 2.d Describe shape of data set: - Data is roughly symmetric since mean = median = mode. - No extreme outliers. - Distribution shape is approximately symmetric and unimodal.