Lecture 2. Diana Pell. Graphical Summaries of Data

Transcription

1 Lecture 2 Diana Pell Graphical Summaries of Data Categorical Frequency Distributions Exercise 1. Twenty-five army inductees were given a blood test to determine their blood type. The data set is Figure 1: Construct a frequency distribution for the data. 1

2 Frequency Distributions for Quantitative Data Exercise 2. These data represent the record high temperatures in degrees Fahrenheit for each of the 50 states. Figure 2: a) Construct a frequency distribution using a class width of 5, and using 100 as the lower class limit for the first class. b) We can put the information in graphical form by constructing a histogram. Construct a frequency histogram from the frequency distribution in part (a). 2

3 c) Construct a relative frequency distribution using the same class width and lower limit for the first class. d) Construct a relative frequency histogram. 3

4 Exercise 3. How much air pollution is caused by motor vehicles? This question was addressed in a study by Dr. Janet Yanowitz at the Colorado School of Mines. She studied the emissions of particulate matter, a form of pollution consisting of tiny particles, that has been associated with respiratory disease. The emissions for 65 vehicles, in units of grams of particles per gallon of fuel, are presented below 2.7.png 2.bb Figure 3: a) Construct a frequency distribution for the data in the table above, using a class width of 1.00 and using 0.00 as the lower class limit for the first class. b) Construct a frequency histogram from the frequency distribution in part (a). 4

5 c) Construct a relative frequency distribution using the same class width and lower limit for the first class. d) Construct a relative frequency histogram. 5

6 Exercise 4. Following is a frequency distribution that presents the number of live births to women aged in the state of Wyoming in a recent year. Figure 4: a) List the lower class limits. b) What is the class width? c) Construct a frequency histogram. 6

7 d) Construct a relative frequency distribution. e) Construct a relative frequency histogram. 7

8 Shapes of Histograms A histogram is symmetric if its right half is a mirror image of its left half. Very few histograms are perfectly symmetric, but many are approximately symmetric. A histogram is skewed if one side, or tail, is longer than the other. A histogram with a long right-hand tail is said to be skewed to the right, or positively skewed. A histogram with a long left-hand tail is said to be skewed to the left, or negatively skewed. Figure 5: (a) A histogram skewed to the left. (b) An approximately symmetric histogram. (c) A histogram skewed to the right. Exercise 5. Classify each of the following histograms as skewed to the left, skewed to the right, or approximately symmetric. Figure 6: Figure 7: 8

9 Exercise 6. Geysers: The geyser Old Faithful in Yellowstone National Park alternates periods of eruption, which typically last from 1.5 to 4 minutes, with periods of dormancy, which are considerably longer. The following table presents the durations, in minutes, of 60 dormancy periods that occurred during the period January 21?25, png 26.bb Figure 8: a) Construct a frequency distribution using a class width of 5, and using 55 as the lower class limit for the first class. b) Construct a frequency histogram from the frequency distribution in part (a). 9

10 c) Construct a relative frequency distribution using the same class width and lower limit for the first class. d) Construct a relative frequency histogram. e) Are the histograms skewed to the left, skewed to the right, or approximately symmetric? 10

11 f) Repeat parts (a) - (d), using a class width of 10, and using 50 as the lower class limit for the first class. g) Do you think that class widths of 5 and 10 are both reasonably good choices for these data, or do you think that one choice is much better than the other? Explain your reasoning. 11