Question

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.

Short answer

The explanatory variable is Duration and the response variable is Distance. The trend in the scatterplot is positive. After calculating, assume the correlation coefficient is near to 1, which shows a strong positive correlation. Assume the equation of the regression line is \( y = mx + c \). The slope of the line \( m \) shows that for every additional second of dance, the distance increases by \( m \) meters. Substituting 1 and 3 seconds in the equation we will get the predicted distance.

Step-by-step solution

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.

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.

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.

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.

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.

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.