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Chapter 3: Problem 6

One of nature's patterns connects the percent of adult birds in a colony that return from the previous year and the number of new adults that join the colony. Here are data for 13 colonies of sparrowhawks: $$\begin{array}{lrrrrrrrrrrrrr}\hline \text { Percent return: } & 74 & 66 & 81 & 52 & 73 & 62 & 52 & 45 & 62 & 46 & 60 & 46 & 38 \\\\\text { New adults: } & 5 & 6 & 8 & 11 & 12 & 15 & 16 & 17 & 18 & 18 & 19 & 20 & 20 \\\\\hline\end{array}$$ Make a scatterplot by hand that shows how the number of new adults relates to the percent of retuming birds.

Short Answer

Expert verified
Plot each (Percent return, New adults) pair on a scatterplot to visualize their relationship.

Step by step solution

01

Understand the Data Points

To create a scatterplot, recognize that each pair of values given represents a point on the plot. The 'Percent return' values are your x-coordinates, and the 'New adults' values are your y-coordinates. Therefore, for each colony, the point will be (Percent return, New adults).
02

Set Up the Axes

Draw two perpendicular lines to represent the x-axis and y-axis. Label the x-axis as 'Percent return' and the y-axis as 'New adults'. Choose an appropriate scale for each axis. Here, you might set the x-axis to range from 0 to 90 (to include all given percentages) and the y-axis from 0 to 25 (to include all new adult numbers).
03

Plot the Points

Using the pair values given, plot each of the 13 points on your graph at the intersection of the x-coordinate and y-coordinate. For example, the first point is (74, 5), the second is (66, 6), and so forth. Keep these points as distinct marks on the plot.
04

Analyze the Scatterplot

Once all points are plotted, examine the pattern formed on the scatterplot. Typically, you look for trends or correlations in the data. For example, do the points form a linear pattern, or is it more scattered? This step involves visually understanding the relationship between the percent return and new adults.

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Key Concepts

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

data visualization
Creating a scatterplot is an essential skill in data visualization, enabling you to observe relationships between two variables effectively. In our scenario, we want to visualize how the number of new sparrowhawk adults relates to the percent of adults returning to their colony. Begin by setting up your axes: the x-axis represents the 'Percent return', and the y-axis represents the 'New adults'. Be sure to choose an appropriate scale that includes all data points. Once the axes are set, plot each pair of data points. This will help you view the relationship spatially, making it easier to spot patterns or trends. The scatterplot is crucial because it offers a clear, visual summary of the data, allowing anyone to interpret the results quickly.
statistical analysis
When you have your scatterplot ready, it opens doors to perform various statistical analyses. Though it seems simple, each plotted point provides insight into your data's distribution and potential trends. The arrangement of points might suggest linearity, clusters, or an overall spread. Statistical analysis might involve calculating the mean of each variable to understand the central tendency or the variance that could imply how spread out the data points are. While the initial analysis relies heavily on visual cues from the scatterplot, you can delve deeper with statistical measures to validate and quantify what you visually perceive. Statistical analysis connects numbers with reality, offering deeper insights into nature's patterns.
correlation study
One of the primary objectives of using a scatterplot in a correlation study is to identify any potential relationship between two variables. In our case, we're interested in seeing how the percentage of returning sparrowhawks might influence the influx of new adults to a colony. By examining the scatterplot, one might look for trends like upward or downward slopes. If the points seem to form an ascending line, this indicates a positive correlation, meaning higher percent returns are associated with more new adults. Conversely, if it appears descending, there may be a negative correlation. Besides just visual inspection, statistical methods like calculating the correlation coefficient can offer a numerical measure of the relationship, verifying the visual patterns observed in the data. This approach aids in understanding the nature of the connection, preparing the ground for more sophisticated predictive modeling.

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Most popular questions from this chapter

Researchers studying acid rain measured the acidity of precipitation in a Colorado wilderness area for 150 consecutive weeks. Acidity is measured by pH. Lower \(\mathrm{pH}\) values show higher acidity. The researchers observed a linear pattern over time. They reported that the regression line \(\widehat{\mathrm{pH}}=5.43-\) \(0.0053(\) week \()\) fit the data well. (a) Identify the slope of the line and explain what it means in this setting. (b) Identify the \(y\) intercept of the line and explain what it means in this setting. (c) According to the regression line, what was the \(\mathrm{pH}\) at the end of this study?

Here are some hypothetical data: $$\begin{array}{lllllll}\hline x & 1 & 2 & 3 & 4 & 10 & 10 \\\y: & 1 & 3 & 3 & 5 & 1 & 11 \\\\\hline\end{array}$$ (a) Make a scatterplot to show the relationship between \(x\) and \(y\) (b) Calculate the correlation for these data by hand or using technology. (c) What is responsible for reducing the correlation to the value in part (b) despite a strong straight-line relationship between \(x\) and \(y\) in most of the observations?

Which of the following is not a characteristic of the least-squares regression line? (a) The slope of the least-squares regression line is always between -1 and 1 (b) The least-squares regression line always goes through the point \((\bar{x}, \bar{y})\) (c) The least-squares regression line minimizes the sum of squared residuals. (d) The slope of the least-squares regression line will always have the same sign as the correlation. (e) The least-squares regression line is not resistant to outliers.

Ninth-grade students at the Webb Schools go on a backpacking trip each fall. Students are divided into hiking groups of size 8 by selecting names from a hat. Before leaving, students and their backpacks are weighed. The data here are from one hiking group in a recent year. Make a scatterplot by hand that shows how backpack weight relates to body weight. $$\begin{array}{lrrrrrrrr}\hline \text { Body weight (?): } & 120 & 187 & 109 & 103 & 131 & 165 & 158 & 116 \\\\\text { Backpack weight (Ib): } & 26 & 30 & 26 & 24 & 29 & 35 & 31 & 28 \\ \hline\end{array}$$

Exercise 6 (page 159 ) examined the relationship between the number of new birds \(y\) and percent of returning birds \(x\) for 13 sparrowhawk colonies. Here are the data once again. $$\begin{array}{lrrrrrrrrrrrrr}\hline \text { Percent return: } & 74 & 66 & 81 & 52 & 73 & 62 & 52 & 45 & 62 & 46 & 60 & 46 & 38 \\\\\text { New adults: } & 5 & 6 & 8 & 11 & 12 & 15 & 16 & 17 & 18 & 18 & 19 & 20 & 20 \\\\\hline \end{array}$$ (a) Use your calculator to help make a scatterplot. (b) Use your calculator's regression function to find the equation of the least-squares regression line. Add this line to your scatterplot from (a). (c) Explain in words what the slope of the regression line tells us. (d) Calculate and interpret the residual for the colony that had \(52 \%\) of the sparrowhawks return and 11 new adults.
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