1. Identify each variable as quantitative or qualitative. 1. a) Amount of time it takes to assemble a simple puzzle is a quantitative variable because it measures a numerical value (time). 1. b) Number of students in a first grade class is a quantitative variable because it counts a numerical quantity. 1. c) Rating of a newly elected politician (excellent, good, fair, poor) is a qualitative variable because it describes categories or qualities. 1. d) State in which a person lives is a qualitative variable because it identifies categories (states). 2. Construct a stem and leaf plot for the heart rate data: Data: 87, 109, 79, 80, 96, 95, 90, 92, 96, 98, 101, 91, 78, 112, 94, 98, 94, 107, 81, 96 Step 1: Sort data in ascending order: 78, 79, 80, 81, 87, 90, 91, 92, 94, 94, 95, 96, 96, 96, 97 (not in data, ignore), 98, 98, 101, 107, 109, 112 Step 2: Identify stems (tens and hundreds place) and leaves (ones place): - 7 | 8 9 - 8 | 0 1 7 - 9 | 0 1 2 4 4 5 6 6 6 8 8 - 10 | 1 7 9 - 11 | 2 Stem and leaf plot: 7 | 8 9 8 | 0 1 7 9 | 0 1 2 4 4 5 6 6 6 8 8 10 | 1 7 9 11 | 2 3. For the set of numbers 6.5, 8.5, 4.7, 9.4, 11.3, 8.5, 9.4, 9.7: Step 1: Calculate the range: $$\text{Range} = \max - \min = 11.3 - 4.7 = 6.6$$ Step 2: Calculate the mean: $$\text{Mean} = \frac{6.5 + 8.5 + 4.7 + 9.4 + 11.3 + 8.5 + 9.4 + 9.7}{8} = \frac{67.9}{8} = 8.49$$ Step 3: Calculate the median: Sort data: 4.7, 6.5, 8.5, 8.5, 9.4, 9.4, 9.7, 11.3 Median is average of 4th and 5th values: $$\text{Median} = \frac{8.5 + 9.4}{2} = 8.95$$ Step 4: Calculate the mean absolute deviation (MAD): Calculate absolute deviations from mean 8.49: |6.5 - 8.49| = 1.99 |8.5 - 8.49| = 0.01 |4.7 - 8.49| = 3.79 |9.4 - 8.49| = 0.91 |11.3 - 8.49| = 2.81 |8.5 - 8.49| = 0.01 |9.4 - 8.49| = 0.91 |9.7 - 8.49| = 1.21 Sum of absolute deviations: $$1.99 + 0.01 + 3.79 + 0.91 + 2.81 + 0.01 + 0.91 + 1.21 = 11.64$$ MAD: $$\frac{11.64}{8} = 1.46$$ Step 5: Calculate the standard deviation: Calculate squared deviations from mean 8.49: $$(6.5 - 8.49)^2 = 3.96$$ $$(8.5 - 8.49)^2 = 0.0001$$ $$(4.7 - 8.49)^2 = 14.36$$ $$(9.4 - 8.49)^2 = 0.83$$ $$(11.3 - 8.49)^2 = 7.91$$ $$(8.5 - 8.49)^2 = 0.0001$$ $$(9.4 - 8.49)^2 = 0.83$$ $$(9.7 - 8.49)^2 = 1.46$$ Sum of squared deviations: $$3.96 + 0.0001 + 14.36 + 0.83 + 7.91 + 0.0001 + 0.83 + 1.46 = 29.39$$ Variance (sample variance with denominator 7): $$s^2 = \frac{29.39}{7} = 4.20$$ Standard deviation: $$s = \sqrt{4.20} = 2.05$$ Final answers: - Range = 6.6 - Mean = 8.49 - Median = 8.95 - Mean absolute deviation = 1.46 - Standard deviation = 2.05