### 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 weโve been given is approximately linear. With that assumption, we can estimate some ๐ฆ-value for a given ๐ฅ-value, and vice versa, by first finding the equation of the regression line. Thatโs ๐ฆ hat equals ๐ plus ๐๐ฅ. Now, ๐ represents the slope of the data. And we can calculate that by finding S๐ฅ๐ฆ divided by S๐ฅ๐ฅ. And thatโs equivalent to ๐ times the sum of ๐ฅ๐ฆ minus the sum of ๐ฅ times the sum of ๐ฆ over ๐ times the sum of ๐ฅ squared minus the sum of all ๐ฅ-values squared. So it follows that to be able to calculate the slope, weโre going to need to find the various sums.

Letโs define the price of a barrel of oil in dollars to be ๐ฅ and the economic growth rate to be ๐ฆ. We can then add two further rows and one further column in our table. Then letโs complete the row that contains our ๐ฅ๐ฆ-values. To do this, we take a ๐ฆ-value and multiply it by its corresponding ๐ฅ-value. So thatโs 26 times 1.8, and thatโs 46.8. Next, we calculate 13.30 times 0.4. Thatโs 5.32. Then, for our third column, itโs 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, weโll find all the ๐ฅ squared values. Remember, ๐ฅ is the price of a barrel of oil in dollars. So our first ๐ฅ 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. Weโll now add up the total of each of our rows. The sum of all the values in our first row, the sum of ๐ฅ, is 180.2. The sum of our ๐ฆ-values is 14.9. Then the total here is the sum of our ๐ฅ๐ฆ-values. Itโs 322.16. Finally, the sum of our ๐ฅ squared values, itโs 4520.08.

With all this in mind, weโre now ready to calculate ๐, the slope of our regression line. Since there are eight pairs of values in our table, ๐ is eight. And so ๐ is eight times 322.16 minus 180.2 times 14.9 over eight times 4520.08 minus the sum of all our ๐ฅ-values squared, so 180.2 squared. Thatโs negative 0.0291.

Next, we need to calculate ๐. And thatโs the value of the ๐ฆ-intercept. We calculate ๐ by finding ๐ฆ bar minus ๐ times ๐ฅ bar, where ๐ฆ bar is the mean of ๐ฆ. Itโs the sum of all the ๐ฆ-values divided by ๐. And ๐ฅ bar is the mean of ๐ฅ. ๐ฆ bar is 14.9 over eight, and ๐ฅ bar is 180.2 over eight. And so that gives us an ๐-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 letโs clear some space and do that. Rounding our values for ๐ and ๐ correct to three significant figures, we get that ๐ฆ hat is 2.52 minus 0.0292๐ฅ. Now, remember, weโre 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 ๐ฅ be equal to 35.40. Thatโs 1.48632. Now, in fact, in our table, each value for economic growth is rounded to one decimal place. So weโll 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.