Regression lesson 4

StarterDangers of Predicting (extrapolation)

Interpretation questionsExam questions

.

Example: The table shows the latitude, x, and mean January temperature(°C), y, for a sample of 10 cities in the northern hemisphere.

Calculate the equation of the regression line of y on x and use it to predict the mean January temperature for the city of Los Angeles, which has a latitude of 34°N.

Regression

City Latitude Mean Jan. temp. (°C)

Belgrade 45 1

Bangkok 14 32

Cairo 30 14

Dublin 50 3

Havana 23 22

Kuala Lumpur 3 27

Madrid 40 5

New York 41 0

Reykjavik 30 –1

Tokyo 36 5

Response variable y

Explanatory variable x

Using a calculator gives A = 33.2713504, B = -0.7202355This is our

estimate of the mean January temperature in

Los Angeles

Regression

So, when x = 34, y = 33.3 – 0.720 × 34 = 8.82°C.

Note: This prediction for the mean January temperature in Los Angeles is based purely on the city’s latitude and takes no account of additional factors that can affect the climate of a city, such as altitude or proximity to the coast

This gives y = 33.3 – 0.720x

Regression

2id

0

5

10

15

20

0 2 4 6 8

d1d2

d3

d4d5

d6

The distances di are referred to as

residuals

Note: The best fitting line should pass through the mean point, .( , )x y

The best fitting line is the one that minimizes the sum of the squared deviations, , where di is the vertical distance between the ith point and the line.

A regression equation can only confidently be used to predict values of y that correspond to x values that lie within the range of the data values available.

Dangers of Predicting (extrapolation)

05

10152025303540

0 5 10 15 20 25

It can be dangerous to extrapolate (i.e. to predict) from the graph, a value for y that corresponds to a value of x that lies beyond the range of the values in the data set.

It is reasonably safe to make

predictions within the range of the

data.

It is unwise to extrapolate beyond the given data.

This is because we cannot be sure that the relationship between the two variables will continue to be true.

Interpretation of A and B for starter

Give an interpretation of A and B in context

Using a calculator gives A = 33.2713504, B = -0.7202355

33.3

Latitude

Meantemperature

A is the temperature when latitude is 0ie the temperature at the equator

B is the decrease in temperature per increase in one degree of latitude.

Interpretation questions from booklet

Exam questions in booklet