Question

find the equation for the least squares regression line of the data described below.
marcy thinks that text messaging is causing her to talk less on the phone. for one month, she examined her text message and call logs with her closest friends.
for each friend, marcy checked the number of text messages she sent to that friend, x, and the number of minutes they spoke on the phone, y.
round your answers to the nearest thousandth.
y = x +

Explanation:

Step1: Calculate necessary sums

First, we need to find the sums of \( x \), \( y \), \( xy \), and \( x^2 \) for the given data points.
The data points are:
\( (x_1,y_1)=(126,232) \), \( (x_2,y_2)=(148,209) \), \( (x_3,y_3)=(285,181) \), \( (x_4,y_4)=(344,162) \), \( (x_5,y_5)=(376,174) \)

Sum of \( x \): \( \sum x = 126 + 148 + 285 + 344 + 376 = 1279 \)

Sum of \( y \): \( \sum y = 232 + 209 + 181 + 162 + 174 = 958 \)

Sum of \( xy \): \( \sum xy = (126\times232)+(148\times209)+(285\times181)+(344\times162)+(376\times174) \)
\( = 29232 + 30932 + 51585 + 55728 + 65424 = 232801 \)

Sum of \( x^2 \): \( \sum x^2 = 126^2 + 148^2 + 285^2 + 344^2 + 376^2 \)
\( = 15876 + 21904 + 81225 + 118336 + 141376 = 378717 \)

Number of data points \( n = 5 \)

Step2: Calculate the slope \( m \)

The formula for the slope \( m \) of the least squares regression line is:
\( m=\frac{n\sum xy - \sum x\sum y}{n\sum x^2 - (\sum x)^2} \)

Substitute the values:
\( m=\frac{5\times232801 - 1279\times958}{5\times378717 - 1279^2} \)

First, calculate the numerator: \( 5\times232801 - 1279\times958 = 1164005 - 1225282 = -61277 \)

Denominator: \( 5\times378717 - 1279^2 = 1893585 - 1635841 = 257744 \)

So, \( m=\frac{-61277}{257744}\approx - 0.2377 \approx -0.238 \) (rounded to nearest thousandth)

Step3: Calculate the y-intercept \( b \)

The formula for the y-intercept \( b \) is:
\( b=\frac{\sum y - m\sum x}{n} \)

Substitute the values:
\( b=\frac{958 - (-0.2377)\times1279}{5} \)
First, calculate \( -0.2377\times1279\approx -304.0283 \), so \( 958 - (-304.0283)=958 + 304.0283 = 1262.0283 \)

Then, \( b=\frac{1262.0283}{5}=252.40566\approx252.406 \) (rounded to nearest thousandth)

Answer:

\( y = -0.238x + 252.406 \)