Question Video: Using Linear Regression Model to Estimate the Value of a Variable at a Certain Point Mathematics

Video Transcript

The table shows the price of a barrel of oil and the economic growth. Using the information in the table, estimate the economic growth if the price of a barrel of oil is 35.40 dollars.

And then we have a table containing eight pairs of entries. So we begin by making an assumption that the bivariate data we’ve been given is approximately linear. With that assumption, we can estimate some 𝑦-value for a given 𝑥-value, and vice versa, by first finding the equation of the regression line. That’s 𝑦 hat equals 𝑎 plus 𝑏𝑥. Now, 𝑏 represents the slope of the data. And we can calculate that by finding S𝑥𝑦 divided by S𝑥𝑥. And that’s equivalent to 𝑛 times the sum of 𝑥𝑦 minus the sum of 𝑥 times the sum of 𝑦 over 𝑛 times the sum of 𝑥 squared minus the sum of all 𝑥-values squared. So it follows that to be able to calculate the slope, we’re going to need to find the various sums.

Let’s define the price of a barrel of oil in dollars to be 𝑥 and the economic growth rate to be 𝑦. We can then add two further rows and one further column in our table. Then let’s complete the row that contains our 𝑥𝑦-values. To do this, we take a 𝑦-value and multiply it by its corresponding 𝑥-value. So that’s 26 times 1.8, and that’s 46.8. Next, we calculate 13.30 times 0.4. That’s 5.32. Then, for our third column, it’s 84.73. Next, we have 28.52. And 26.7 times 3.2 is 85.44. And we continue this row in this manner.

Next, we’ll find all the 𝑥 squared values. Remember, 𝑥 is the price of a barrel of oil in dollars. So our first 𝑥 squared value is 26 squared, which is 676. Next, we calculate 13.3 squared, which is 176.89. Our next value is 524.41. 12.4 squared is 153.76. And we can continue this row in the same way. We’ll now add up the total of each of our rows. The sum of all the values in our first row, the sum of 𝑥, is 180.2. The sum of our 𝑦-values is 14.9. Then the total here is the sum of our 𝑥𝑦-values. It’s 322.16. Finally, the sum of our 𝑥 squared values, it’s 4520.08.

With all this in mind, we’re now ready to calculate 𝑏, the slope of our regression line. Since there are eight pairs of values in our table, 𝑛 is eight. And so 𝑏 is eight times 322.16 minus 180.2 times 14.9 over eight times 4520.08 minus the sum of all our 𝑥-values squared, so 180.2 squared. That’s negative 0.0291.

Next, we need to calculate 𝑎. And that’s the value of the 𝑦-intercept. We calculate 𝑎 by finding 𝑦 bar minus 𝑏 times 𝑥 bar, where 𝑦 bar is the mean of 𝑦. It’s the sum of all the 𝑦-values divided by 𝑛. And 𝑥 bar is the mean of 𝑥. 𝑦 bar is 14.9 over eight, and 𝑥 bar is 180.2 over eight. And so that gives us an 𝑎-value of 14.9 over eight minus negative 0.0291 and so on times 180.2 over eight. And that gives us 2.517 and so on.

We now have everything we need to be able to create the equation of the regression line. So let’s clear some space and do that. Rounding our values for 𝑎 and 𝑏 correct to three significant figures, we get that 𝑦 hat is 2.52 minus 0.0292𝑥. Now, remember, we’re trying to find the economic growth if the price of a barrel of oil is 35.40 dollars. So we can find this by letting 𝑥 be equal to 35.40. That’s 1.48632. Now, in fact, in our table, each value for economic growth is rounded to one decimal place. So we’ll do the same with our value. That gives us 1.5. So the estimate for the economic growth if the price of a barrel of oil is 35.4 dollars is 1.5.