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.

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.

Transcript text: \begin{tabular}{cc} 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 \\ \hline 5 & 9 \\ 62 & 66 \end{tabular} 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 $\square$ more messages received. B. For each additional message received, there is an average of $\square$ 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 $\square$ message(s) received. B. With 0 messages received, there should be about $\square$ message(s) sent.

Solution

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.

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

Step 1: Organize the Data

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 $.

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

Step 2: Calculate the Slope and Intercept

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.

Step 3: Compute Summations

First, we compute the necessary summations:

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

Step 4: Calculate the Slope

Substitute the summations into the slope formula:

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

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