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Chapter 2: Problem 2

Find the best-fitting straight line to the given set of data, using the method of least squares. Graph this straight line on a scatter diagram. Find the correlation coefficient. $$ (0,1),(1,2),(2,2) $$

Short Answer

Expert verified
The best-fitting line is \( y = 0.5x + 1.1667 \) and the correlation coefficient is approximately 0.866.

Step by step solution

01

Setup the equation of a line

We need to find the best-fitting line in the form of \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
02

Calculate values for least squares method

We need to calculate necessary sums: \( \sum x_i \), \( \sum y_i \), \( \sum x_i y_i \) and \( \sum x_i^2 \). For our data:\[ \sum x_i = 0 + 1 + 2 = 3 \]\[ \sum y_i = 1 + 2 + 2 = 5 \]\[ \sum x_i y_i = (0 \times 1) + (1 \times 2) + (2 \times 2) = 6 \]\[ \sum x_i^2 = (0^2) + (1^2) + (2^2) = 5 \].
03

Determine slope (m)

Use the formula \( m = \frac{n\sum(x_i y_i) - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2} \). Plug in values: \( n = 3 \).\[m = \frac{3 \times 6 - 3 \times 5}{3 \times 5 - 3^2} = \frac{18 - 15}{15 - 9} = \frac{3}{6} = 0.5\]
04

Determine the y-intercept (c)

Use the formula \( c = \frac{\sum y_i - m \sum x_i}{n} \). Plug in values:\[c = \frac{5 - 0.5 \times 3}{3} = \frac{5 - 1.5}{3} = \frac{3.5}{3} \approx 1.1667\].
05

Formulate the equation of the line

Now with \( m = 0.5 \) and \( c \approx 1.1667 \), the line equation becomes: \( y = 0.5x + 1.1667 \).
06

Calculate the correlation coefficient (r)

The formula for the correlation coefficient \( r \) is:\[r = \frac{n\sum(x_i y_i) - \sum x_i \sum y_i}{\sqrt{(n\sum x_i^2 - (\sum x_i)^2)(n\sum y_i^2 - (\sum y_i)^2)}}\]We need to calculate \( \sum y_i^2 = 1^2 + 2^2 + 2^2 = 9 \).\[r = \frac{3\times 6 - 3\times 5}{\sqrt{(15 - 9)(27 - 25)}} = \frac{3}{\sqrt{6 \times 2}} = \frac{3}{\sqrt{12}} = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2} \approx 0.866\].
07

Sketch the scatter plot and line of best fit

Plot the points (0,1), (1,2), and (2,2) on a graph. Draw the line \( y = 0.5x + 1.1667 \) through the points to visually represent the best fit.

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Key Concepts

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

Best-Fitting Line
When working with data, finding the best-fitting line, also known as the line of best fit, helps us understand relationships between two data variables. Using the least squares method, we aim to minimize the sum of the squares of the vertical distances between data points and the line. This results in a model that predicts linear relationships most accurately.

In our example exercise, data points were
  • (0, 1)
  • (1, 2)
  • (2, 2)
Using the least squares method, the calculations involved determining the slope (\(m\)) and y-intercept (\(c\)) of the line characterized by \(y = mx + c\). After performing the necessary summations and applying the formulas, the best-fitting line was determined to be \(y = 0.5x + 1.1667\). This line runs through the data in a way that minimizes the overall deviation of all points from the line.

A best-fitting line is critical in predicting values and estimating the strength of relationships in data analysis. It鈥檚 the foundational tool when working with linear trends and forecasts.
Correlation Coefficient
The correlation coefficient, often symbolized as \(r\), measures the strength and direction of a linear relationship between two variables. This value ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 means no linear relationship, and 1 indicates a perfect positive linear relationship.

In this problem, computing \(r\) involves the same sums used for finding the line of best fit, but applies them through a different formula. For the given data, the correlation coefficient was found to be roughly \(r \approx 0.866\), signifying a strong positive relationship. This value tells us that as one variable increases, the other variable tends to increase as well, closely following the linear line of best fit.

A crucial takeaway here is that correlation doesn't imply causation. Even a perfect correlation doesn't mean that one variable causes the other to happen. Correlation coefficients help in determining linear relationships but should be used cautiously in interpreting broader relationships.
Scatter Plot
A scatter plot is a graph that displays data points plotted on a horizontal and vertical axis, showing how much one variable is affected by another. It鈥檚 an essential tool for investigating the relationships between variables. Each point on the scatter plot corresponds to one pair of values from the dataset.

In our example, plotting the points (0, 1), (1, 2), and (2, 2) creates a visual representation. On this graph, we drew the best-fitting line, \(y = 0.5x + 1.1667\), to visualize the trend the data follows. The line that runs through these points helps in seeing the direction and strength of the relationship.

Visualizing data via scatter plots aids in identifying patterns, trends, and outliers. It's a preliminary step before performing further statistical analyses like calculating the best-fitting line or correlation coefficient. Understanding what the scatter plot reveals can guide decision-making and ensure statistical analyses are grounded in accurately represented data.

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Most popular questions from this chapter

Number of Roman Catholic Priests The following table gives the number of Roman Catholic ordinations per year in the United States for selected years and can be found in Glassman. \(^{64}\) $$ \begin{array}{|l|cccccc|} \hline \text { Year } & 1967 & 1972 & 1977 & 1982 & 1987 & 1992 \\ \hline \text { Number } & 932 & 705 & 613 & 453 & 365 & 289 \\ \hline \end{array} $$ a. On the basis of this data, find the best-fitting exponential function using exponential regression. Let \(x=0\) correspond to 1967 . Graph. Use this model to estimate the number in \(1997 .\) b. Using the model in part (a), estimate when the number of Roman Catholic ordinations per year will reach \(150 .\)

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uarez-Villa and Karlsson \(^{47}\) studied the relationship between the sales in Sweden's electronic industry and production costs (per unit value of product sales). Their data are presented in the following table. $$\begin{array}{|c|ccccc|}\hline x & 7 & 13 & 14 & 15 & 17 \\\\\hline y & 0.91 & 0.72 & 0.91 & 0.81 & 0.72 \\\\\hline x & 20 & 25 & 27 & 35 & 45 \\\\\hline y & 0.90 & 0.77 & 0.65 & 0.73 & 0.70 \\\\\hline x & 45 & 63 & 65 & 82 & \\\\\hline y & 0.78 & 0.82 & 0.92 & 0.96 & \\\\\hline\end{array}$$ Here \(x\) is product sales in millions of krona, and \(y\) is production costs per unit value of product sales. a. Use cubic regression to find the best-fitting cubic to the data and the correlation coefficient. Graph. b. Find the minimum production costs.

Economic Entomology Smitley and Davis \(^{69}\) studied the changes in gypsy moth egg mass densities over one generation as a function of the initial egg mass density in a control plot and two treated plots. The data below are for the control plot. $$ \begin{array}{|c|cccc|} \hline \begin{array}{c} \text { Initial Egg Mass } \\ \text { (per 0.04 ha) } \end{array} & 50 & 75 & 100 & 160 \\ \hline \begin{array}{c} \text { Change in Egg Mass } \\ \text { Density (\%) } \end{array} & 250 & -100 & -25 & -25 \\ \hline \begin{array}{c} \text { Initial Egg Mass } \\ \text { (per 0.04 ha) } \end{array} & 175 & 180 & 200 & \\ \hline \begin{array}{c} \text { Change in Egg Mass } \\ \text { Density (\$) } \end{array} & -50 & 50 & 0 \\ \hline \end{array} $$ a. On the basis of the data given in the table, find the bestfitting logarithmic function using least squares. (Note that the authors used logarithms to the base 10.) Graph. b. Use this model to estimate the change in egg mass density with an initial egg mass of 150 per 0.04 ha.

Plant Resistance Talekar and Lin \(^{21}\) collected the data shown in the table that relates the number of trichomes (hairs) per \(6.25 \mathrm{~mm}^{2}\) on pods of soybeans to the percentage of pods damaged by the lima bean pod borer. \begin{tabular}{|c|cccccc|} \hline\(x\) & 208 & 212 & 230 & 255 & 260 & 335 \\ \hline\(y\) & 9 & 9 & 10 & 8 & 14 & 25 \\ b. Find the correlation coefficient. c. Interpret what the slope of the line means. \hline \end{tabular} Here \(x\) is the number of trichomes per \(6.25 \mathrm{~mm}^{2}\) on the pods, and \(y\) is the percentage of damaged pods. a. Use linear regression to find the best-fitting line that relates the number of trichomes on the pods to the percentage of pods damaged.
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