91Ó°ÊÓ

The following table shows the number of text messages sent and received by some people in one day. (Source: StatCrunch: Responses to survey How often do you text? Owner: Webster West. A subset was used.) a. Make a scatterplot of the data, and state the sign of the slope from the scatterplot. Use the number sent as the independent variable. b. Use linear regression to find the equation of the best-fit line. Graph the line with technology or by hand. c. Interpret the slope. d. Interpret the intercept. $$ \begin{aligned} &\begin{array}{|c|c|} \hline \text { Sent } & \text { Received } \\ \hline 1 & 2 \\ \hline 1 & 1 \\ \hline 0 & 0 \\ \hline 5 & 5 \\ \hline 5 & 1 \\ \hline 50 & 75 \\ \hline 6 & 8 \\ \hline 5 & 7 \\ \hline 300 & 300 \\ \hline 30 & 40 \\ \hline \end{array}\\\ &\begin{array}{|r|r|} \hline \text { Sent } & \text { Received } \\ \hline 10 & 10 \\ \hline 3 & 5 \\ \hline 2 & 2 \\ \hline 5 & 5 \\ \hline 0 & 0 \\ \hline 2 & 2 \\ \hline 200 & 200 \\ \hline 1 & 1 \\ \hline 100 & 100 \\ \hline 50 & 50 \\ \hline \end{array} \end{aligned} $$

Step by step solution

01

Plotting a Scatterplot

Plot the sent messages on the x-axis and received messages on the y-axis. The slope sign must be interpreted from the direction of the scatterplot

02

Applying Linear Regression

Apply a linear regression to the scatterplot data, aiming to find the least squares best fit line, the equation of which can be expressed in the form y = mx + c, where m is the slope and c is the intercept.

03

Interpretation of the Slope

The slope represents the change in received messages as per the change in sent messages. If the slope is positive, it means that an increase in sent messages corresponds to an increase in received messages and vice versa.

04

Interpretation of the Intercept

The intercept represents the expected value of received messages when no messages are sent. For example, if the intercept is high, it might suggest that someone receives many messages regardless of whether they send any.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatterplot

A scatterplot is a type of graph used to display the relationship between two different sets of data. To create one, you plot individual data points on a grid where the x-axis represents one variable (in this case, the number of text messages sent) and the y-axis represents another variable (the number of messages received). Each point on the scatterplot corresponds to one pair of data from the table.

The value in a scatterplot comes from its ability to visually express trends and correlations between the two variables. For instance, if the points on your scatterplot tend to rise from left to right, this suggests a positive correlation, meaning as the number of sent messages increases, the number of received messages tends to increase as well. Conversely, if the points fall from left to right, this suggests a negative correlation. In our exercise, by plotting the data, we expect the slope to be positive as a general increase in sent messages will likely correspond to an increase in received messages.

Slope Interpretation

In the context of linear regression, the slope of the line (\(m\) in the equation) is pivotal as it indicates the strength and direction of the relationship between the two variables being analyzed. When interpreting the slope, you're looking at the amount by which the dependent variable (received messages) changes for a one-unit increase in the independent variable (sent messages).

If the slope is positive, as we've seen with our scatterplot, for every extra message sent, there is an associated increase in the number of messages received. The steepness of the slope tells you how significant this change is. A steeper slope indicates a stronger relationship, implying that sending additional messages has a more substantial effect on the number of messages received.

Intercept Interpretation

The intercept (\(c\) in the equation), sometimes called the y-intercept, is the point where the regression line crosses the y-axis. It's the value of the dependent variable when the independent variable is zero. In the context of our texting data, the intercept tells us the expected number of received messages when no messages are sent.

This value can offer insights into baseline levels of communication. For instance, a high intercept could imply that a person receives a lot of messages without needing to send any out. Conversely, an intercept close to zero suggests that the number of received messages is heavily dependent on the number of sent messages.

Least Squares Method

The least squares method is a standard approach to the fitting of a linear regression line to a set of data points, minimizing the sum of the squares of the differences between the observed dependent variable values and those predicted by the linear function. Essentially, it finds the best-fit line that reduces the overall error.

By applying the least squares method to our texting data, we determine the most accurate linear representation of the relationship between the number of sent and received messages. This process yields the slope and intercept values that we can use to predict communication patterns.

Quantitative Data Analysis

Quantitative data analysis involves the application of statistical methods to understand the properties and trends within numerical data. In our exercise, we've used quantitative data analysis to establish a relationship (via linear regression) between two quantitative variables: the number of messages sent and received.

This analysis helps us reveal patterns and quantify correlations, allowing us to make predictions based on numerical evidence. Once the relationship is modeled, we can test new data points against our model to predict outcomes, which is invaluable for individuals and businesses aiming to understand behavior or trends.