Question Video: Calculating a Least Squares Regression Model from a Sum of Squares | Nagwa Question Video: Calculating a Least Squares Regression Model from a Sum of Squares | Nagwa

Question Video: Calculating a Least Squares Regression Model from a Sum of Squares Mathematics • Third Year of Secondary School

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For a given data set, π‘₯ bar = 30.8, 𝑦 bar = 27.5, 𝑆_(π‘₯π‘₯) = 4,407, 𝑆_(𝑦𝑦) = 228, and 𝑆_(π‘₯𝑦) = 752. Find the equation of the regression line of 𝑦 on π‘₯ in the form 𝑦 = π‘šπ‘₯ + 𝑏, giving π‘š and 𝑏 correct to two decimal places.

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Video Transcript

For a given data set, π‘₯ bar equals 30.8, 𝑦 bar equals 27.5, 𝑆 π‘₯π‘₯ equals 4,407, 𝑆 𝑦𝑦 equals 228, and 𝑆 π‘₯𝑦 equals 752. Find the equation of the regression line of 𝑦 on π‘₯ in the form 𝑦 equals π‘šπ‘₯ plus 𝑏, giving π‘š and 𝑏 correct to two decimal places.

The equation that describes the least squares regression line of 𝑦 on π‘₯ is 𝐲 hat equals π‘Ž plus 𝑏π‘₯. We’ll compare this with the general form 𝑦 equals π‘šπ‘₯ plus 𝑏 at the end. In the equation for 𝐲 hat equals π‘Ž plus 𝑏π‘₯, 𝑏 is the slope. And it’s found by dividing 𝑆 π‘₯𝑦 by 𝑆 π‘₯π‘₯. Then, once we have a value for 𝑏, we can find a value for π‘Ž, which is the 𝑦-intercept, by calculating 𝑦 bar minus 𝑏 times π‘₯ bar.

So we begin by calculating the value of 𝑏. We’re given 𝑆 π‘₯𝑦 is equal to 752 and 𝑆 π‘₯π‘₯ is 4,407. So 𝑏 is 752 divided by 4,407. That’s 0.17063 and so on, which correct to two decimal places is 0.17.

Now that we have a value for 𝑏, we can calculate π‘Ž. 𝑦 bar is 27.5. We just calculated 𝑏, and we’re going to use the exact value for 𝑏. And π‘₯ bar is 30.8. So π‘Ž is 27.5 minus 752 over 4,407 times 30.8, which is 22.2443 and so on. Correct to two decimal places, that’s 22.24.

Substituting these back into the equation 𝐲 hat equals π‘Ž plus 𝑏π‘₯, and we get 𝐲 hat equals 22.24 plus 0.17π‘₯. Now, we’ve been in fact been told to give this in the form 𝑦 equals π‘šπ‘₯ plus 𝑏. So we switch the π‘₯-term and the constant. 𝑦 equals 0.17π‘₯ plus 22.24.

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