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5.2 – Linear regression

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Definition: Simple linear regression is used to analyze relationships between two variables.

Two Main Objectives:

  1. Identify Relationships – Determine if a statistically significant relationship exists between two variables (positive or negative correlation).
  2. Make Predictions – Use the relationship to forecast new observations.

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Establish if there is a relationship between two variables.

  • More specifically, establish if there is a statistically significant relationship between the two.

ANALYSE AND WRITE RELATIONSHIP BETWEEN THE FOLLOWING :

Examples:

Income and spending :

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wage and gender:

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student height and exam scores.

We can use regression models to test this

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Forecast new observations.

  • Can we use what we know about the relationship to forecast unobserved values?

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  • Examples:
  • What will sales be over the next quarter?

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  • What will the ROI of a new store opening be contingent on store attributes

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When is a Regression Line Suitable?

 

 

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English mark

Maths mark

 

The smaller the total error, the better the straight line fits the data, and the more appropriate the linear model is to represent the data.

Model

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When is a Regression Line unreliable?

 

 

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Hours of revision

Test score

Model

“We should not use the model to make predictions about hours revised, because score depends on hours revised.”

Here the hours of revision is the explanatory/independent variable, and the test score the response/dependent variable. In general, use of the model should only be used to estimate a value of the response variable.

Comment on this statement.

I can use my regression line to predict to predict the number of hours someone has revised when they score 70% in the test.

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When is a Regression Line unreliable?

 

 

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Hours of revision

Test score

Model

“10 hours is outside the range of the data, so the estimate would be unreliable.”

 

Comment on this statement.

interpolation

extrapolation

extrapolation

I can use my regression line to predict to predict someone’s test score if they’ve done 10 hours of revision.

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How do we interpret the gradient of 3?

 

 

 

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How do we interpret the gradient of 3?

 

 

 

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Regression line

Interpretation of gradient

Interpretation of y-intercept

Hours revised h and maths mark m

m=40+10h

For each extra hour revised, students achieve 10 marks better on average.

A student doing no revision would expect to score 40 marks.

Daily average temperature t (in °C) and daily sales of coffee s (in $)

s=5000-150t

Extra increase in temperature of 1°C results in an average decrease in sales of $150

Sales would be $5000 when the temperature is at freezing point (0°C)

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Quickfire

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Regression line

Interpretation of gradient

Interpretation of y-intercept

Hours revised h and maths mark m

m=40+10h

For each extra hour revised, students achieve 10 marks better on average.

A student doing no revision would expect to score 40 marks.

Daily average temperature t (in °C) and daily sales of coffee s (in $)

s=5000-150t

Extra increase in temperature of 1°C results in an average decrease in sales of $150

Sales would be $5000 when the temperature is at freezing point (0°C)

a

b

Quickfire

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