Question Video: Interpreting Linear Regression Coefficients in Context Mathematics

The latitude (π‘₯) and the average temperatures in February (𝑦, measured in Β°C) of 10 world cities were measured. The calculated least squares linear regression model for this data was 𝑦 = 35.7 βˆ’ 0.713π‘₯. What is the interpretation of the value of βˆ’0.713 in the model? And what is the interpretation of the value of 35.7 in the model?

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Video Transcript

The latitude π‘₯ and the average temperatures in February 𝑦 measured in degrees Celsius of 10 world cities were measured. The calculated least squares linear regression model for this data was 𝑦 equals 35.7 minus 0.713π‘₯. What is the interpretation of the value of negative 0.713 in the model? And what is the interpretation of the value of 35.7 in the model?

Let’s begin by taking a look at the equation that we have for the least squares linear regression line. The least squares regression line is essentially a way of modeling the line of best fit for the data. In this case then, our line of best fit has the equation 𝑦 equals 35.7 minus 0.713π‘₯. Comparing this to the general formula for the equation of a straight line and we see that the 𝑦-intercept for our line is 35.7. Then the gradient or the slope is the coefficient of π‘₯; it’s negative 0.713.

So, what are π‘₯ and 𝑦? We’re told that π‘₯ is the latitude of the world city and 𝑦 is the average temperature in February measured in degrees Celsius. And we know that negative 0.713 is the slope of the line. But what does that mean when we link latitude and the average temperature in February of the world city? Well, a slope of negative 0.713 means that when we move one unit right along our line, we also move 0.713 units down. Since π‘₯ is the latitude and the average temperature is 𝑦, this means that, for each additional degree of latitude, the average temperature decreased by 0.713 degrees Celsius.

Now, we’ll move on to the second part of this question, and this asks us to interpret the value of 35.7. Remember, 35.7 is the 𝑦-intercept here. And the 𝑦-intercept is the value of 𝑦 when π‘₯ is equal to zero. π‘₯ being equal to zero means that we’re looking at the temperature in February for a city of latitude zero or a city on the equator.

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