Question

Cynthia has measured the weight and miles per gallon of four different cars, listed the data in a table, and graphed the results on a scatterplot. She noticed the points fall closely on a line. Weight, in hundreds of poundsMiles per Gallon (mpg) 532 1027 1225 1522 Using the data values that Cynthia collected, select the correct slope and y-intercept. a.) s l o p e equals short dash 1 semicolon y minus i n t e r c e p t equals 37 b.) s l o p e equals 1 semicolon y minus i n t e r c e p t equals 27 c.) s l o p e equals 1 semicolon y minus i n t e r c e p t equals short dash 27 d.) s l o p e equals short dash 1 semicolon y minus i n t e r c e p t equals 0

Answer 1

Explanation:

To find the slope (m) and y-intercept (b) of the linear relationship between weight and miles per gallon (mpg) from the given data, you can use the formula for the equation of a straight line:

y = mx + b

where:

y is the dependent variable (mpg in this case)

x is the independent variable (weight in hundreds of pounds)

m is the slope (the rate of change)

b is the y-intercept (the value of y when x = 0)

Now, let's calculate the values:

Using the given data points:

Weight (x) and Miles per Gallon (mpg) are as follows:

(5, 32), (10, 27), (12, 25), (15, 22)

First, calculate the slope (m):

m = (Σxy - nΣxΣy) / (Σx^2 - n(Σx)^2)

where Σ represents summation (sum), n is the number of data points, x is weight, and y is mpg.

n = 4

Σx = 5 + 10 + 12 + 15 = 42

Σy = 32 + 27 + 25 + 22 = 106

Σxy = (5 * 32) + (10 * 27) + (12 * 25) + (15 * 22) = 160 + 270 + 300 + 330 = 1060

Σx^2 = (5^2) + (10^2) + (12^2) + (15^2) = 25 + 100 + 144 + 225 = 494

Now, plug these values into the formula:

m = (1060 - 4 * 42 * 106) / (494 - 4 * (42)^2)

m = (1060 - 4 * 42 * 106) / (494 - 4 * 1764)

m = (1060 - 17712) / (494 - 7056)

m = (-16652) / (-6562)

m ≈ 2.53 (rounded to two decimal places)

Now that we have the slope (m), we can find the y-intercept (b) using one of the data points, for example, (5, 32):

32 = 2.53 * 5 + b

32 = 12.65 + b

Now, solve for b:

b = 32 - 12.65

b ≈ 19.35

So, the equation relating weight (in hundreds of pounds) and miles per gallon (mpg) is approximately:

mpg ≈ 2.53 * weight (in hundreds of pounds) + 19.35

This corresponds to option (b):

slope (m) equals 1;

y-intercept (b) equals 19.35.