Question

The scatter plot shows the number of text messages sent (x) and text messages received (y) by10 students in a half hour.



A scatter plot.Short description, A scatter plot.,Long description,
There are ten points on the scatter plot in a generally upward trend. They have the coordinates listed.

(0, 2).
(5, 1).
(6, 4).
(8, 12).
(11, 11).
(12, 13).
(15, 11).
(17, 20).
(21, 22).
(24, 23).

Question
Which equation could be used to find the ,begin emphasis,best,end emphasis, estimate for the number of text messages received based on the number of text messages sent?

Answer 1

To find the best estimate for the number of text messages received based on the number of text messages sent, we can use linear regression. The equation for a linear regression line is y = mx + b, where y is the dependent variable (number of text messages received), x is the independent variable (number of text messages sent), m is the slope of the line, and b is the y-intercept.

To find the best estimate for the slope (m) and y-intercept (b), we can use a set of data points. From the given scatter plot, we have the following coordinates:

(0, 2).
(5, 1).
(6, 4).
(8, 12).
(11, 11).
(12, 13).
(15, 11).
(17, 20).
(21, 22).
(24, 23).

We can use these points to calculate the slope (m) and y-intercept (b) using the least squares method.

Calculating the slope (m):
m = (Σxy - (Σx)(Σy)/n) / (Σx^2 - (Σx)^2 / n)
where Σxy is the sum of the product of x and y values,
Σx is the sum of x values,
Σy is the sum of y values,
and n is the number of data points.

Calculating the y-intercept (b):
b = (Σy - m(Σx)) / n

Using the given data points, the slope (m) and y-intercept (b) can be calculated as follows:

Σxy = (0)(2) + (5)(1) + (6)(4) + (8)(12) + (11)(11) + (12)(13) + (15)(11) + (17)(20) + (21)(22) + (24)(23) = 1547
Σx = 0 + 5 + 6 + 8 + 11 + 12 + 15 + 17 + 21 + 24 = 119
Σy = 2 + 1 + 4 + 12 + 11 + 13 + 11 + 20 + 22 + 23 = 119
Σx^2 = (0^2) + (5^2) + (6^2) + (8^2) + (11^2) + (12^2) + (15^2) + (17^2) + (21^2) + (24^2) = 1111

n = 10

Using these values, we can calculate the slope (m) and y-intercept (b):

m = (1547 - (119)(119)/10) / (1111 - (119)^2/10)
= (1547 - 1409.9) / (1111 - 1416.1)
= 137.1 / (-305.1)
≈ -0.449

b = (119 - (-0.449)(119)) / 10
≈ 1.66

Therefore, the equation that could be used to find the best estimate for the number of text messages received based on the number of text messages sent is:

y = -0.449x + 1.66