Mathematics 1000, Winter 2008

Transcription

1 Mathematics 1000, Winter 2008 Lecture 4 Sheng Zhang Department of Mathematics Wayne State University January 16, 2008

2 Announcement Monday is Martin Luther King Day NO CLASS

3 Today s Topics Curves and Histograms 1 Curves and Histograms Scatterplots

4 From histograms to curves A histogram consists of several rectangles, each of the same width, all based on a line. The height of each rectangle represents the number of data points in a given range. For large data sets, we can have many rectangles, each quite narrow. If they are narrow enough, the histogram smoothes out, and the top of it resembles a curve, not merely a jagged line.

5 Number of finishers per hour, New York Marathon, :59 3 3:59 4 4:59 5 5:59 6 6:59 7 7:59 8 8:59

6 Number of finishers per half-hour, New York Marathon :30 3 3:30 4 4:30 5 5:30 6 6:30 7 7:30 8

7 3000 Number of finishers per ten minutes New York Marathon, :09 2:10 2:19 2:20 2:29 2:30 2:39 3 3:09 3:30 3:39 4 4:09 4:30 4:39 5 5:09 5:30 5:39 6 6:09 6:30 6:39 7 7:09 7:30 7:39 8 8:09

8 New York Marathon results, five minute by five minute histogram.

9 Per capita GDP by country in 115 poor countries Number of countries Per capita GDP in dollars

10 Per capita GDP by country in 115 poor countries Number of countries Per capita GDP in thousands of dollars

11 Number of countries Per capita GDP by country in 115 poor countries Per capita GDP in hundreds of dollars

12 Per capita GDP by country in 115 poor countries Number of countries Per capita GDP in hundreds of dollars

13 Interpreting curves When a distribution is given by a curve, the rectangles have vanished. The areas below the curve represent the proportion of the data that lie within a given horizontal range.

14 Special typse of curves With a large number of data points, distributions tend to resemble curves. There are many possible curves that can serve as models for how the data should lie. We will only consider one of them.

15 Example of a normal distribution

16 Properties of the normal distribution symmetric mean and median are the same quartiles lie about 2/3 of a standard deviation from the mean satisfies the rule

17 The Rule In a normal distribution with mean x and standard deviation s: 68% of the observations lie between x s and x + s 95% of the observations lie between x 2s and x + 2s 99.7% of the observations lie between x 3s and x + 3s

18 Application Curves and Histograms ACT scores approximately follow a normal distribution with mean 20.8 and standard deviation 4.8. This means that 68% of such scores lie between 16 ( = ) and 25.6 ( = ). So 32% lie outside that range. Since the distribution is symmetric, roughly half of the 32% will lie below 16. That is, 16% of scores will lie below 16.

19 Exercise Curves and Histograms If adult women s heights are distributed normally with a mean of 64.5 inches and a standard deviation of 2.5 inches, what proportion of women will be under 59.5 inches tall? Answer: 59.5 inches is 2 standard deviations below the mean. So 5% ( = 100% - 95%) of women will have height farther from the mean than that. Half of them will be short, and half will be tall. So 2.5% will be shorter than 59.5 inches tall.

20 Exercise Curves and Histograms If adult women s heights are distributed normally with a mean of 64.5 inches and a standard deviation of 2.5 inches, what proportion of women will be under 59.5 inches tall? Answer: 59.5 inches is 2 standard deviations below the mean. So 5% ( = 100% - 95%) of women will have height farther from the mean than that. Half of them will be short, and half will be tall. So 2.5% will be shorter than 59.5 inches tall.

21 WSU application If there are 10,000 women at this university, how many would we expect to be under 59.5 inches tall? Answer: About 250, 2.5% of 10,000.

22 WSU application If there are 10,000 women at this university, how many would we expect to be under 59.5 inches tall? Answer: About 250, 2.5% of 10,000.

23 Calculators Curves and Histograms The Chapter 6 material will require the calculator more intensively than the material we covered up to now. If you need help with using the calculator, then you should be sure to get the one that we support. The quiz instructors and the tutors in the Mathematics Resource Center may well be able to help you with calculators.

24 Scatterplots Some applications of two-variable statistics Do students who spend more time studying get better grades? Do people who smoke tend to die earlier? What is the relationship between the amount of carbon dioxide in the atmosphere and the average global temperature?

25 Two variable statistics Scatterplots One variable statistics: Study shape, center, spread, look for outliers. Step 1: Draw a picture. Two variable statistics: Study patterns, relationship (correlation and regression), look for outliers. Step 1: Draw a picture.

26 Two variable statistics Scatterplots One variable statistics: Study shape, center, spread, look for outliers. Step 1: Draw a picture. Two variable statistics: Study patterns, relationship (correlation and regression), look for outliers. Step 1: Draw a picture.

27 Overview of next four lectures Scatterplots Scatterplots (today) Regression lines Correlation and regression Interpretation

28 Example 1 from the text Scatterplots Student Beers BAC Student Beers BAC

29 Scatterplots We plot one variable on the horizontal axis, another on the vertical axis. The variable on the horizontal axis is called the explanatory variable. The variable on the variable axis is called the response variable. Key question: How much does the explanatory variable explain the response variable?

30 Scatterplots Beer and Blood Alcohol 0.20 Blood alcohol content Beers

31 Scatterplots Beer and Blood Alcohol 0.20 Blood alcohol content Beers

32 Scatterplots State versus national standards There is a national test to measure the proficiency of fourth graders in mathematics. Most students are not proficient in mathematics according to this measure. Each of the states has its own separate proficiency test. Let s compare how students did on their two tests.

33 Scatterplots State versus national standards

34 Smoking and mortality Scatterplots A much debated example: Does smoking cause health problems, or do people likely to have bad health just tend to smoke more?

35 Scatterplots Smoking and mortality scatterplot Mortality rate from coronary heart disease and number of cigarettes smoked per day 375 Death rate year old males Cigarettes per day Source: 1969 Surgeon General s Report