The least-squares regression line is:
a. the line that makes the sum of the squares of the vertical distances of the data points from the line (the sum of squared residuals) as small as possible.
b. the line that passes through the most data points.
c. All of the answer options are correct.
d. the line such that half of the data points fall above the line and half fall below the line.

a. the line that makes the sum of the squares of the vertical distances of the data points from the line (the sum of squared residuals) as small as possible.

Step-by-step explanation:

If we have N points and we want to adjust a model

We can define the error associated to this like that:

So as we can see here we are adding the square distances between the real and the adjusted values in order to minimize the error for this reason the correct answer is:

a. the line that makes the sum of the squares of the vertical distances of the data points from the line (the sum of squared residuals) as small as possible.

For this case we need to calculate the slope with the following formula:

Where:

And we can find the intercept using this:

step-by-step :

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt