MAT Mathematics in Today's World

Transcription

1 MAT 1000 Mathematics in Today's World

2 Last Time We learned how to calculate correlation (abbreviated r). This is a number that tells us about the strength and direction of an association, when that association has the form of a straight line. If the sign of the number r is positive, the association has a positive direction, if r is negative, so is the association. The closer r is to either 1 or -1, the stronger the association, and this means that a scatterplot of the data will look more like a straight line.

3 Today First: a warning about interpreting correlation. We will also talk about least-squares regression. This is a way to calculate the line that is the best fit for the data, in other words: a line that is a good approximation of the scatterplot. The reason least-squares regression is important is that it allows us to make predictions where we don t have any data these predictions will be based on the pattern the data gives us.

4 Correlation is not causation You may have heard this expression before. What does it mean? Correlation is good evidence for a cause and effect relationship between two variables. If there is such a relationship, the variables will have a strong correlation. On the other hand, variables can have a strong correlation even though there is no cause and effect relationship.

5 Correlation is not causation Example Ice cream sales are correlated with drowning deaths. Obviously not a cause and effect relationship. In this case the explanation is that ice cream sales and drowning deaths are both related to the weather. More ice cream is sold in the summer, and more people go swimming in the summer. We call this relationship between ice cream sales and drowning deaths mutual response.

6 Correlation is not causation Correlation may not even be due to mutual response. Example (The Pirate Effect) The number of pirates is correlated with global average temperature: over the past few centuries the number of pirates has decreased, and global average temperatures have increased. Is global warming caused by lack of pirates? This is just a coincidence. People call this kind of relationship a nonsense correlation. For more nonsense correlations:

7 Approximating scatterplots Last time we calculated the correlation between the heights and weights of five male adults. Here is that same data as a scatterplot.

8 Approximating scatterplots If you had to draw by hand a line that approximated the shape of this scatterplot, you could end up with any number of lines.

9 Approximating scatterplots For example, maybe you would draw this line

10 Or this one Approximating scatterplots

11 Approximating scatterplots But there is only one least-squares regression line:

12 Review of linear functions The goal is to take a set of pairs of data and produce a line that approximates that data. First, we need to review some facts about lines. In mathematics we describe a line using a linear function. Linear functions can be put into a special form, called slopeintercept form. This looks like: y = m x + b

13 Review of linear functions In the equation y = m x + b, the numbers m and b are called constants. This just means that they should have specific values. For example y = 2 x + 3 is the equation of a line. Here we specify that m = 2 and b = 3. We never specify x or y both of these are variables. We usually call the number m the slope of the line, and b is called the intercept.

14 Review of linear functions But how does the equation describe a line? y = 2 x + 3 There are many pairs of numbers x and y that satisfy this equation. For example, the pair x = 1 and y = 5. How can we tell? Plug in 1 for x and 5 for y: This is a true equation. 5 = 2 (1) + 3

15 Review of linear functions Of course, not every pair of numbers satisfies the equation y = 2 x + 3 For example, the pair x = 1 and y = 2 Try plugging in 1 for x and 2 for y: 2 = 2 (1) + 3 This is not a true equation.

16 Review of linear functions To find the graph of the line described by the equation y = 2 x + 3, we need two pairs x and y that satisfy the equation. We have one, namely the pair x = 1 and y = 5, let s find one more. What we can do is pick any number for x and solve for y. Let s say x = 3. Plug that into the equation: y = If you simplify this, you will find that y must be equal to 9.

17 Review of linear functions Now we use these pairs to plot two points. The number x gives the horizontal location of the point, the number y gives the vertical distance.

18 Review of linear functions Plot the pair x = 1 and y = 5

19 Review of linear functions Add in the pair x = 3 and y = 9

20 Review of linear functions Now connect these two points with a line

21 The least-squares regression line A line y = m x + b will be determined by knowing the values of m and b. We will give formulas for finding each of these. In the equation y = m x + b we have two variables: x is the explanatory variable y is the response variable

22 The least-squares regression line Remember that our starting point in all of this is a collection of paired data. Therefore we have two variables. We also have their means x and y, and their standard deviations s x and s y. We can also calculate their correlation r. The formulas for the least-square regression line use all of these numbers.

23 The least-squares regression line The slope of the least-squares regression line is The intercept is m = r s y s x b = y m x Notice the m in the equation for the intercept this is the slope m (the same one we find with the previous formula).

24 The least-squares regression line Example Last time we considered the heights and weight of five adult males. We found the following numbers: x = 71.8 inches y = 197 pounds r = 0.9 So the least-squares regression line has slope m = = 7.73 The intercept is b = s x = 3.96 inches s y = pounds 71.8 = 358

25 The least-squares regression line Example Putting these together, the least-squares regression line is: y = 7.73 x 358

26 The least-squares regression line Note that none of the data actually lies on the line. For a line to be the least-squares regression line the distance from all of the data to the line must be as small as possible. Nevertheless, the line need not (and usually does not) contain any of the data values.

27 Predictions The most important application of least-squares regression lines is for making predictions. If a scatterplot has a linear form, this suggests an underlying pattern. Mathematically, that pattern is exactly the least-squares regression line. We can then make predictions based on the pattern we see in the data we ve collected.

28 Predictions Let s use our least-squares regression line y = 7.73 x 358 to make predictions. What does our data predict will be the weight of a man who is 70 inches tall? To find this, we plug 70 in for x and find the corresponding y. y = = pounds How about a man who is 76 inches tall? y = = pounds

29 Predictions

30 Predictions In general, when you have paired data, you can find the leastsquares regression line y = m x + b The variable x always corresponds to the explanatory variable (if there is one), and y is always the response variable. You can use the regression line to predict values of the response variable for different values of the explanatory variable. Just take the value of the explanatory variable, and substitute it for x in the equation. The number your get for y is the predicted response.

31 Predictions One danger in using least-squares regression for predictions is extrapolation. Within the range of our data, the least-squares regression line should give reasonable predictions. But, if we plug in numbers too far outside that range, the predictions may no longer be reasonable. In our original height and weight data, the heights range from 67 inches to 77 inches. We can be confident that our least-squares regression line gives reasonable predictions for any height in this range.

32 Predictions What weight does our regression line y = 7.73 x 358 predict for a man who is 5 feet tall (60 inches)? pounds. This is quite low: even 120 pounds is considered a low weight for a 5 foot tall man. What about 50 inches? 28.5 pounds. This is obviously preposterous.