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Chapter 2: Problem 7

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. $$ (0,4),(1,2),(2,2),(3,1),(4,1) $$

Short Answer

Expert verified
The best-fit line is \(y = -\frac{2}{5}x + 2.8\) with correlation coefficient \(r \approx -0.854\).

Step by step solution

01

Organize Data Points

List the given data points as pairs of \((x, y)\) coordinates: \((0,4), (1,2), (2,2), (3,1), (4,1)\). Identify the values of \(x\) and \(y\) for each observation. This organization helps in calculating sums necessary for later steps.
02

Calculate Sums

Compute the following sums which are needed for the least squares equations: \( \, \sum x = 0+1+2+3+4 = 10, \, \sum y = 4+2+2+1+1 = 10, \, \sum xy = 0\times4 + 1\times2 + 2\times2 + 3\times1 + 4\times1 = 16, \, \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 30.\) The number of data points, \(n\), is 5.
03

Determine Line Equation Coefficients (Least Squares)

Use the formulas for the slope \(m\) and the intercept \(b\) of the best-fit line: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] and \[ b = \frac{(\sum y) - m(\sum x)}{n}. \] Substitute the values to find: \(m = \frac{5\times16 - 10\times10}{5\times30 - 10^2} = \frac{80 - 100}{150 - 100} = \frac{-20}{50} = -\frac{2}{5}\), and \(b = \frac{10 - (-\frac{2}{5})\times10}{5} = \frac{10 + 4}{5} = \frac{14}{5} = 2.8\). The line equation is \(y = -\frac{2}{5}x + 2.8\).
04

Calculate the Correlation Coefficient

The correlation coefficient \(r\) is given by: \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}. \] Compute \(\sum y^2 = 4^2 + 2^2 + 2^2 + 1^2 + 1^2 = 26\). Substitute the values: \( r = \frac{5\times16 - 10\times10}{\sqrt{(150)(130) - (100)(100)}} = \frac{-20}{\sqrt{150\times30} -100} \approx -0.854 \).
05

Graph the Best-Fit Line and Data Points

Plot the points \((0,4), (1,2), (2,2), (3,1), (4,1)\) on a scatter plot. Draw the line \(y = -\frac{2}{5}x + 2.8\) using the slope-intercept form. Start at the y-intercept \(2.8\) and use the slope \(-\frac{2}{5}\) (down 2 units, right 5 units) to determine another point on the line and extend the line through these points.

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Key Concepts

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

Linear Regression
Linear regression is a statistical technique used to find the best-fitting straight line through a set of data points. This method primarily focuses on modeling the relationship between a dependent variable (often represented as "y") and an independent variable ("x") by minimizing the sum of the squared differences between the observed and predicted values.
In the context of least squares linear regression, you compute two key components to define the line: the slope ("m") and the y-intercept ("b").
  • The slope indicates the rate of change or the steepness of the line.
  • The y-intercept is the point where the line intersects the y-axis.
This linear model is particularly useful for predicting values and understanding the relationship between variables. The linear equation derived from this method is typically represented as: \[ y = mx + b \] where "m" is the slope and "b" is the y-intercept.
Correlation Coefficient
The correlation coefficient, denoted as "r", measures the strength and direction of a linear relationship between two variables in your dataset.
Its value ranges from -1 to 1.
  • An "r" value near 1 indicates a strong positive correlation: as one variable increases, the other also increases.
  • An "r" value near -1 indicates a strong negative correlation: as one variable increases, the other decreases.
  • An "r" value around 0 indicates little or no linear correlation between the variables.
In the formula used to calculate "r," consider factors like the sums of products of paired scores, the sum of squared scores, and the number of data points ("n"). A substantial r-value, either positive or negative, implies a reliable linear relationship, which can be critical for effective data analysis.
Scatter Plot
A scatter plot is a visual representation of the relationship between two numerical variables. In a scatter plot, individual data points are plotted in a coordinate system, allowing you to identify potential patterns or correlations at a glance.
Scatter plots are fundamental tools in data analysis because they:
  • Provide an immediate visual context for observing trends or patterns.
  • Allow for a visual estimation of the line of best fit, which can then be quantified through methods like linear regression.
  • Help in identifying any outliers that may affect the analysis.
When analyzing data using a scatter plot, you will often look for clusters or trends that indicate a linear relationship, which can later be mathematically translated into a regression line to model the relationship between the variables.
Data Analysis
Data analysis involves systematically applying logical or statistical techniques to evaluate data and extract meaningful insights. This process encompasses everything from collecting and organizing data to applying various analytical methods such as linear regression.
Key steps in data analysis include:
  • Data Collection: Gathering accurate and relevant data, a critical first step for analysis.
  • Data Organization: Structuring the data in a logical format, like a table of \(x, y\) pairs, to facilitate further analysis.
  • Descriptive Analysis: Summarizing the main features of the data, often using visual tools like scatter plots or summary statistics.
  • Inferential Analysis: Using statistical methods to draw conclusions and make predictions from data.
Performing comprehensive data analysis can reveal relationships between variables, like identifying how well a linear model fits the collected data, and lead to informed decision-making based on statistical evidence.

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Most popular questions from this chapter

Economies of Scale in Plant Size Strategic Management \(^{8}\) relates a study in economies of scale in the machine tool industry. The data is found in the following table. \begin{tabular}{|c|ccccccc|} \hline\(x\) & 70 & 115 & 130 & 190 & 195 & 400 & 450 \\ \hline\(y\) & 1.1 & 1.0 & 0.85 & 0.75 & 0.85 & 0.67 & 0.50 \\ \hline \end{tabular} Here \(x\) is the plant capacity in thousands of units, and \(y\) is the employee- hours per unit. a. Determine the best-fitting line using least squares and the correlation coefficient. b. Is there an advantage in having a large plant? Explain. c. What does this model predict the employee-hours per unit will be when the plant capacity is 300,000 units? d. What does this model predict the plant capacity will be if the employee- hours per unit is \(0.90 ?\)

VCR Sales The following table gives the sales of VCRs in the United States in millions for some selected years. \(^{72}\) a. On the basis of the data given for the years 1978 to 1988 , find the best- fitting exponential function using exponential regression. Determine the correlation coefficient. Graph. Using this model, estimate sales in \(1992 .\) b. Now find the best-fitting logistic curve. Graph. Using this model, estimate sales in \(1992 .\) Note that the actual factory sales in 1992 was 66.78 million. $$ \begin{array}{|l|cccccc|} \hline \text { Year } & 1978 & 1980 & 1982 & 1984 & 1986 & 1988 \\ \hline \text { Sales } & 0.20 & 0.84 & 2.53 & 8.88 & 30.92 & 51.39 \\ \hline \end{array} $$

Moose Reproductive Effort Ericsson and colleagues \(^{31}\) studied the effect of the age of a female moose on the mortality of her offspring. They collected data shown in the table relating the age of the female moose to offspring mortality during the hunting season. $$ \begin{array}{|l|cccc|} \hline \text { Moose age } & 2 & 3 & 4 & 5 \\\ \hline \text { Mortality of Offspring } & 0.5 & 0.4 & 0.25 & 0.35 \\ \hline \text { Moose age } & 6 & 7 & 8 & 9 \\ \hline \text { Mortality of Offspring } & 0.35 & 0.5 & 0.37 & 0.35 \\ \hline \text { Moose age } & 10 & 11 & 12 & 13 \\\ \hline \text { Mortality of Offspring } & 0.48 & 0.37 & 0.53 & 0.38 \\\ \hline \text { Moose age } & 14 & & & \\ \hline \text { Mortality of Offspring } & 0.60 & & & \\ \hline \end{array} $$ a. Find the best-fitting quadratic (as the researches did) relating age to mortality of offspring and the square of the correlation coefficient. b. Find the age at which mortality of offspring is minimized.

Productivity Bernstein \({ }^{13}\) studied the correlation between productivity growth and gross national product (GNP) growth of six countries: France (F), Germany (G), Italy (I), Japan (J), the United Kingdom (UK), and the United States (US). Productivity is given as output per employeehour in manufacturing. The data they collected for the years \(1950-1977\) is given in the following table. \begin{tabular}{|c|cccccc|} \hline Country & US & UK & F & I & G & J \\ \hline\(x\) & 2.5 & 2.7 & 5.2 & 5.6 & 5.7 & 9.0 \\ \hline\(y\) & 3.5 & 2.3 & 4.9 & 4.9 & 5.7 & 8.5 \\ \hline \end{tabular} Here \(x\) is the productivity growth \((\%),\) and \(y\) is the GNP growth (\%). a. Determine the best-fitting line using least squares and the correlation coefficient. b. What does this model predict the GNP growth will be when the productivity growth is \(7 \% ?\) c. What does this model predict the productivity growth will be if the GNP growth is \(7 \% ?\)

Grace and colleagues \(^{53}\) found a correlation between the percent increase in the individual weight of foraging workers and the percent decrease in colony population during the latter stages of the life of the termite colony. Their data are found in the following table. Use power regression to find the best-fitting power function to the data and the correlation coefficient. Graph. $$\begin{array}{|c|cccc|} \hline x & 7 & 32 & 45 & 120 \\\\\hline y & 50 & 62 & 73 & 79 \\\\\hline\end{array}$$ . Here \(x\) is the percent increase in the weight of individual foraging workers in millimeters, and \(y\) is the percent decrease in the population of the colony.
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