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Chapter 12: Problem 20

Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where \(x\) is the number of endorsements the player has and \(y\) is the amount of money made (in millions of dollars). $$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\\ \hline\end{array}$$ Table 12.13 Draw a scatter plot of the data.

Short Answer

Expert verified
Plot each point from the data table on a graph to create the scatter plot.

Step by step solution

01

Understand the Data

We have a table with two columns: \(x\), which represents the number of endorsements, and \(y\), which represents the amount of money made (in millions of dollars). The table provides ten pairs of \((x, y)\) values that we will plot on a graph.
02

Plot the Points

Use the coordinate plane to plot each pair of \((x, y)\) from the table. For example, the first pair is \((0, 2)\), so place a point at 0 on the x-axis and 2 on the y-axis. Continue this for each of the ten points: \((0,2), (3,8), (2,7), (1,3), (5,13), (4,10), (5,12), (4,9), (3,9), (0,3)\).
03

Draw the Scatter Plot

Once all points are plotted, the scatter plot will visually represent the relationship between \(x\) and \(y\). Look for any patterns, trends, or outliers. The x-axis represents the number of endorsements, and the y-axis represents the amount of money made.

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Key Concepts

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

Coordinate Plane
A coordinate plane is a two-dimensional surface where you can plot points, lines, and curves. It's like a map, with one line going left and right, called the x-axis, and another going up and down, known as the y-axis. These axes allow you to graphically represent data by plotting points at intersections of these lines.

The coordinate plane is divided into four areas called quadrants. Each point on the coordinate plane has a pair of numerical coordinates written as \(x, y\), where \(x\) is the position along the x-axis and \(y\) is the position along the y-axis. This setup helps in visualizing relationships between two variables. In our exercise, athletes' endorsements and their earnings are matched on this coordinate plane.
Data Points
Data points are individual sets of values represented on a coordinate plane. In our exercise, each data point comes from the table provided. These points are noted as ordered pairs, such as \(0, 2\), where \(0\) is the number of endorsements (x-coordinate) and \(2\) is the amount of money made (y-coordinate).

To plot a data point on a scatter plot, locate the x-coordinate on the x-axis and the corresponding y-coordinate on the y-axis, and place a dot where these align. By doing this with all the data points, you'll get a visual display of how the values relate to each other. Looking at these plotted points can reveal patterns and trends in the data, helping us better understand the relationship between the number of endorsements athletes have and the money they make.
X-Axis and Y-Axis
The x-axis and y-axis form the backbone of any coordinate plane, offering a logical structure for data visualization.

  • X-Axis: It's the horizontal axis of the coordinate plane. In our scenario, it measures the number of endorsements. Each unit along this line represents an increment of one endorsement.
  • Y-Axis: This is the vertical axis, representing the amount of money made, measured in millions of dollars. Each unit here corresponds to one million dollars.
By marking where values intersect on these axes, you construct data points that help us understand how the variables relate. Positive trends, for example, might show up as an upward diagonal, indicating that more endorsements result in higher earnings. Using these axes effectively is key in analyzing and interpreting the data in scatter plots.

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Most popular questions from this chapter

Ornithologists, scientists who study birds, tag sparrow hawks in 13 different colonies to study their population. They gather data for the percent of new sparrow hawks in each colony and the percent of those that have returned from migration. Percent return:74; 66; 81; 52; 73; 62; 52; 45; 62; 46; 60; 46; 38 Percent new:5; 6; 8; 11; 12; 15; 16; 17; 18; 18; 19; 20; 20 a. Enter the data into your calculator and make a scatter plot. b. Use your calculator鈥檚 regression function to find the equation of the least-squares regression line. Add this to your scatter plot from part a. c. Explain in words what the slope and y-intercept of the regression line tell us. d. How well does the regression line fit the data? Explain your response. e. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain. f. An ecologist wants to predict how many birds will join another colony of sparrow hawks to which 70% of the adults from the previous year have returned. What is the prediction?

If the level of significance is 0.05 and the \(p\) -value is \(0.04,\) what conclusion can you draw?

The maximum discount value of the Entertainment庐 card for the 鈥淔ine Dining鈥� section, Edition ten, for various pages is given in Table 12.21 $$\begin{array}{|c|c|}\hline \text { Page number } & {\text { Maximum value (s) }} \\ \hline 4 & {16} \\ \hline 14 & {19} \\ \hline 25 & {19} \\\ \hline 25 & {17} \\ \hline 43 & {17} \\ \hline 42 & {15} \\ \hline 72 & {15} \\ \hline 85 & {17} \\ \hline 90 & {17} \\ \hline\end{array}$$ a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the ordered pairs. c. Calculate the least-squares line. Put the equation in the form of: \(\hat{y}=a+b x\) d. Find the correlation coefficient. Is it significant? e. Find the estimated maximum values for the restaurants on page ten and on page 70 . f. Does it appear that the restaurants giving the maximum value are placed in the beginning of the 鈥淔ine Dining鈥� section? How did you arrive at your answer? g. Suppose that there were 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200? h. Is the least squares line valid for page 200? Why or why not? i. What is the slope of the least-squares (best-fit) line? Interpret the slope.

If there are 15 data points in a set of data, what is the number of degree of freedom?

Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where \(x\) is the number of endorsements the player has and \(y\) is the amount of money made (in millions of dollars). $$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\\ \hline\end{array}$$ Table 12.13 Use regression to find the equation for the line of best fit.
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