Examples

Air Temperature

The maximum and minimum daily air temperature was recorded at Paisley, Glasgow, over the last 50 years. These temperatures are displayed on the scatterplot below.

Based on this plot, what can we say about the relationship between maximum and minimum air temperature?

Observations may be:

• You can see that as the minimum temperature increases, the maximum temperature also increases.
  
• This suggests there could be a linear relationship between the maximum and minimum temperatures.
  
• This means that we could draw a straight line through these data points that captures the nature of this relationship.

Recall the equation of a straight line \[y=mx+c.\] In regression modelling we assume

Typically, in regression modelling, we use Greek letter to denote the gradient and y-interpret of the line. In particular, we could write this mode as \[y=\alpha + \beta x.\]

Given that we typically have several observations, for instance in this example we have 50 observations of max and min air temperature. We would use a subscript to indicate observation

We could write this mode as \[y_i=\alpha + \beta x_i \hspace{0.5cm} i=1,\ldots,n.\]

Power

A study of an individual’s power (measured by a vertical jump and converted to power using the Lewis formula) and its relationship to their weight was undertaken by a sports scientist. A random sample of 38 users of the Stevenson Building facilities was selected and their power and weight measured.

Is there a linear relationship between power and weight?

Which curve do you think best describes this relationship?

Notice here that D is not a straight line. Of course straight lines are not always the best way to describe relationships. Assuming C, try writing down the intercept and slope of this line. We will revisit this example later.

Alcohol consumed and blood alcohol content

In a study of alcohol consumption and related blood alcohol content, 16 student volunteers at Ohio State University drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their percent blood alcohol content (BAC). The researcher is interested in finding out if the number of cans of beers influences the BAC measurement.

In this example we see that both the variables are numeric and the relationship between the variable BAC and the number of cans of beers appear to be have a positive relationship.

Do you think the black line in the second plot is the best fitted linear relationship between these two variables?