Questions: Sent Received 2 6 0 1 78 81 67 69 150 152 3 3 225 230 325 328 5 8 4 6 0 4 2 5 10 13 3 7 6 8 6 7 6 7 3 4 5 9 62 66 c. Interpret the slope. Select the correct choice below and fill in the answer box to complete your choice. (Round to three decimal places as needed.) A. For each additional message sent, there is an average of more messages received. B. For each additional message received, there is an average of more messages sent. d. Interpret the intercept and comment on it. Select the correct choice below and fill in the answer box to complete your choice. (Round to the nearest whole number as needed.) A. With 0 messages sent, there should be about message(s) received. B. With 0 messages received, there should be about message(s) sent.

Solution Steps

To solve this problem, we need to perform a linear regression analysis on the given data to find the slope and intercept of the best-fit line. This will allow us to interpret the relationship between the number of messages sent and received.

1. Extract the data from the table.
2. Use a linear regression model to find the slope and intercept.
3. Interpret the slope and intercept based on the regression results.

We are given a table of messages sent and received. Let's denote the number of messages sent as \( x \) and the number of messages received as \( y \).

\[ \begin{array}{cc} \text{Sent} & \text{Received} \\ 2 & 6 \\ 0 & 1 \\ 78 & 81 \\ 67 & 69 \\ 150 & 152 \\ 3 & 3 \\ 225 & 230 \\ 325 & 328 \\ 5 & 8 \\ 4 & 6 \\ 0 & 4 \\ 2 & 5 \\ 10 & 13 \\ 3 & 7 \\ 6 & 8 \\ 6 & 7 \\ 6 & 7 \\ 3 & 4 \\ 5 & 9 \\ 62 & 66 \\ \end{array} \]

To find the slope (\( m \)) and intercept (\( b \)) of the linear regression line \( y = mx + b \), we use the formulas:

\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]

\[ b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2} \]

Where \( n \) is the number of data points.

First, we compute the necessary summations:

\[ \sum x = 957, \quad \sum y = 1008, \quad \sum x^2 = 204,073, \quad \sum y^2 = 213,204, \quad \sum xy = 214,073 \]

Substitute the summations into the slope formula:

\[ m = \frac{20(214,073) - (957)(1008)}{20(204,073) - (957)^2} \]

\[ m = \frac{4,281,460 - 964,656}{4,081,460 - 915,849} \]

\[ m = \frac{3,316,804}{3,165,611} \approx 1.0478 \]

Substitute the summations into the intercept formula:

\[ b = \frac{(1008)(204,073) - (957)(214,073)}{20(204,073) - (957)^2} \]

\[ b = \frac{205,716,984 - 204,872,961}{4,081,460 - 915,849} \]

\[ b = \frac{844,023}{3,165,611} \approx 0.2666 \]

The slope \( m \approx 1.0478 \) indicates that for each additional message sent, there is an average of 1.048 more messages received.

The intercept \( b \approx 0.2666 \) indicates that with 0 messages sent, there should be about 0.267 messages received.

Final Answer