91Ó°ÊÓ

When honeybee scouts find a food source or a nice site for a new home, they communicate the location to the rest of the swarm by doing a "waggle dance." 74 They point in the direction of the site and dance longer for sites farther away. The rest of the bees use the duration of the dance to predict distance to the site. Table 2.32 Duration of \(a\) honeybee waggle dance to indicate distance to the source $$\begin{array}{cc} \hline \text { Distance } & \text { Duration } \\ \hline 200 & 0.40 \\\250 & 0.45 \\ 500 & 0.95 \\\950 & 1.30 \\ 1950 & 2.00 \\\3500 & 3.10 \\\4300 & 4.10 \\\\\hline\end{array}$$ Table 2.32 shows the distance, in meters, and the duration of the dance, in seconds, for seven honeybee scouts. \(^{75}\) This information is also given in HoneybeeWaggle. (a) Which is the explanatory variable? Which is the response variable? (b) Figure 2.70 shows a scatterplot of the data. Does there appear to be a linear trend in the data? If so, is it positive or negative? (c) Use technology to find the correlation between the two variables. (d) Use technology to find the regression line to predict distance from duration. (e) Interpret the slope of the line in context. (f) Predict the distance to the site if a honeybee does a waggle dance lasting 1 second. Lasting 3 seconds.

Step by step solution

01

Identify explanatory and response variables

The explanatory variable is Duration of the dance since it explains or causes the change in the response variable. The response variable is Distance because the distance is influenced by how long the bees dance.

02

Analyze the trend in the scatterplot

By plotting all the points from the table in a scatterplot and observing the general trend, it appears that there is a positive linear correlation. It indicates as the duration of the dance increases, the distance to the location also increases.

03

Find the correlation

Use technology (such as calculator or statistical software) to calculate the Pearson Correlation Coefficient. A positive value closer to 1 indicates a strong positive correlation.

04

Find the regression line

Use technology to find the equation of the regression line. The equation of the regression line will be in the form of \( y = mx + c \), where \( y \) is the distance, \( x \) is the duration, \( m \) is the slope, and \( c \) is the y-intercept.

05

Interpret the slope of the line

The slope of the line represents the change in the distance for every additional second the bee dances. This means if the slope is \( m \), then dancing an additional second increases the distance by \( m \) meters.

06

Predict the distance

To predict the distance, substitute the duration (1 second and 3 seconds) into the regression line equation. The resulting value predicts the distance to the site based on the duration of the waggle dance.

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Key Concepts

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

Explanatory Variable

In the study of linear regression, the explanatory variable plays a critical role. It is the variable that is considered to cause or explain changes in another variable, known as the response variable. In our example of the honeybee waggle dance, the duration of the dance is the explanatory variable. This is because it is assumed that the length of the dance provides information about, and potentially influences, the distance the bees must travel to reach the food source or new home site.
By identifying the explanatory variable, researchers can investigate how it affects the response variable. This helps in understanding the relationship between the two variables, which is crucial in building a predictive model. When you are trying to determine which is your explanatory variable, a good question to ask is: which variable is providing information or is doing the explaining?

Response Variable

The response variable in a linear regression analysis is the outcome that the researcher is trying to predict or explain. It relies on the changes introduced by the explanatory variable. In the honeybee example, the response variable is the distance that the bees travel. This distance is influenced by the duration of the waggle dance, which indicates its dependency on the explanatory variable.
Understanding which variable is the response helps in constructing a model to predict it using the explanatory variable. This allows researchers to interpret how changes in the explanatory variable affect the response variable. By analyzing the response variable, we gain insights into how phenomena of interest occur and potentially control or replicate these outcomes in further studies.

Scatterplot Analysis

Scatterplot analysis is an essential tool in understanding the relationship between two quantitative variables. It graphically represents the data points on a two-dimensional plot, with each axis representing one of the variables.
This visualization helps detect patterns, trends, and potential correlations between the variables. For the honeybee data, plotting duration on the x-axis and distance on the y-axis lets us see the positive linear trend, indicating that as the waggle dance duration increases, so does the distance.
By examining scatterplots, you can identify whether a relationship exists and determine its directionality---whether positive, negative, or nonexistent. It helps in deciding whether a linear model is suitable and how strong the correlation might be. Thus, scatterplot analysis is vital before proceeding to deeper statistical analysis such as correlation or regression.