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Chapter 13: Problem 69

A population data set produced the following information. $$ \begin{aligned} &N=460, \quad \Sigma x=3920, \quad \Sigma y=2650, \quad \Sigma x y=26,570, \\ &\Sigma x^{2}=48,530, \text { and } \Sigma y^{2}=39,347 \end{aligned} $$ Find the linear correlation coefficient \(\rho\).

Short Answer

Expert verified
The linear correlation coefficient \(\rho\) is the value that you obtain after substituting the given values into the formula and completing the computations. Use a calculator for this task.

Step by step solution

01

Understand the correlation coefficient formula

Firstly, it's important to know that the formula for the population correlation coefficient is: \[ \rho = \frac{N(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[N(\Sigma x^2) - (\Sigma x)^2][N(\Sigma y^2) - (\Sigma y)^2]}} \] where \(N\) represents the number of pairs of scores, each set of parenthesis represent a sum over all observations.
02

Substitute the given values into the numerator

The numerator of the correlation coefficient formula is \[N(\Sigma xy) - (\Sigma x)(\Sigma y)\] Substitute the given values into this part of the formula, the numerator will be \[460*26570 - 3920*2650\]
03

Substitute values into the denominator

The denominator of the correlation coefficient formula is \[\sqrt{[N(\Sigma x^2) - (\Sigma x)^2][N(\Sigma y^2) - (\Sigma y)^2]}\] Substituting the given values into this part of the formula, the denominator will be \[sqrt{(460*48530 - 3920^2)(460*39347 - 2650^2)}\]
04

Complete the Computations

Use a calculator to simplify both the numerator and the denominator, then divide the numerator by the denominator to find the linear correlation coefficient \(\rho\).

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Key Concepts

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

Population Dataset
A population dataset refers to the complete set of data or observations collected from every member of a particular group that you're studying. In the problem provided, we have a population dataset consisting of several summary values derived from 460 pairs of observations (represented by the variable \(N\)).

The key figures we have include:
  • \( \Sigma x = 3920 \): the sum of all \(x\)-values.
  • \( \Sigma y = 2650 \): the sum of all \(y\)-values.
  • \( \Sigma xy = 26,570 \): the sum of the product of each pair of \(x\) and \(y\) values.
  • \( \Sigma x^2 = 48,530 \): the sum of the squares of \(x\)-values.
  • \( \Sigma y^2 = 39,347 \): the sum of the squares of \(y\)-values.
These calculations are necessary to evaluate the strength and direction of the relationship between the \(x\) and \(y\) variables using statistical methods like calculating the linear correlation coefficient.

This entire dataset will help us eventually derive \(\rho\), which measures the linear association between the two variables.
Numerator and Denominator
In mathematical terms, the numerator and denominator are the two fundamental parts that comprise a fraction. For the linear correlation coefficient formula, these parts have specific roles.

The numerically computed relationship between \(x\) and \(y\) is housed within:
  • **Numerator:** Expression showing co-variation, i.e., \(N(\Sigma xy) - (\Sigma x)(\Sigma y)\). This component uncovers how much the x and y values vary together.
  • **Denominator:** This part normalizes the relationship by incorporating variance. It's expressed as \(\sqrt{[N(\Sigma x^2) - (\Sigma x)^2][N(\Sigma y^2) - (\Sigma y)^2]}\). This evaluates how the x and y values vary independently.

The correlation coefficient itself depends on both components. As the values in these two parts influence \(\rho\), a thorough substitution of values is compulsory to achieve accuracy.
Substitution Method
The substitution method involves the systematic replacement of variables with their corresponding numerical values. For this problem, we substitute values into the formula for calculating the linear correlation coefficient, \(\rho\).

Here's a structured way of doing it:
  • **Numerator Substitution:** Insert the given values into the formula \(460(26,570) - 3920(2650)\).
  • **Denominator Substitution:** Insert into the expression \(\sqrt{(460*48,530 - 3920^2)(460*39,347 - 2650^2)}\).

Once the appropriate values are in place, the method demands careful computation to ensure each calculation step is executed correctly. This stands crucial for subsequent simplification and accuracy.
Computation Steps
After substituting the values comes the computation steps, where the aim is to break down the mathematical processes into manageable pieces. Here's how you handle it:
  • Calculate the **numerator**: Engage a calculator for precise multiplication and subtraction of \(460*26570 - 3920*2650\).
  • Evaluate the **denominator**: Resolve each squared term and square root within \((460*48,530 - 3920^2)(460*39,347 - 2650^2)\).
  • Divide the formulated **numerator** by the **denominator**: This gives the linear correlation coefficient \(\rho\).

By breaking down these steps thoroughly, one can achieve a comprehensive understanding of deriving \(\rho\). This becomes vital to determining the linear association's strength and direction within your dataset.

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Most popular questions from this chapter

Construct a \(95 \%\) confidence interval for the mean value of \(y\) and a \(95 \%\) prediction interval for the predicted value of \(y\) for the following. a. \(\hat{y}=13.40+2.58 x\) for \(x=8\) given \(s_{e}=1.29, \bar{x}=11.30, \mathrm{SS}_{x x}=210.45\), and \(n=12\) b. \(\hat{y}=-8.6+3.72 x\) for \(x=24\) given \(s_{e}=1.89, \bar{x}=19.70, \mathrm{SS}_{x x}=315.40\), and \(n=10\)

Explain the meaning of independent and dependent variables for a regression model.

A sample data set produced the following information. $$ \begin{aligned} &n=10, \quad \Sigma x=100, \quad \Sigma y=220, \quad \Sigma x y=3680, \\ &\Sigma x^{2}=1140, \text { and } \Sigma y^{2}=25,272 \end{aligned} $$ a. Calculate the linear correlation coefficient \(r\). b. Using a \(2 \%\) significance level, can you conclude that \(\rho\) is different from zero?

A population data set produced the following information. $$ \begin{aligned} &N=250, \quad \Sigma x=9880, \quad \Sigma y=1456, \quad \Sigma x y=85,080, \\ &\Sigma x^{2}=485,870, \text { and } \Sigma y^{2}=135,675 \end{aligned} $$ Find the linear correlation coefficient \(\rho\).

The following table gives information on the limited tread warranties (in thousands of miles) and the prices of 12 randomly selected tires at a national tire retailer as of July 2012 . $$ \begin{array}{l|rrrrrrrrrrrr} \hline \text { Warranty (thousands of miles) } & 60 & 70 & 75 & 50 & 80 & 55 & 65 & 65 & 70 & 65 & 60 & 65 \\ \hline \text { Price per tire (\$) } & 95 & 135 & 94 & 90 & 121 & 70 & 140 & 80 & 92 & 125 & 160 & 155 \\ \hline \end{array} $$ a. Taking warranty length as an independent variable and price per tire as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y y}\) a. Taking warranty length as an independent variable and price per tire as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Find the regression of price per tire on warranty length. c. Briefly explain the meaning of the values of \(a\) and \(b\) calculated in part b. d. Calculate \(r\) and \(r^{2}\) and explain what they mean. e. Plot the scatter diagram and the regression line. f. Predict the price of a tire with a warranty length of 73,000 miles. g. Compute the standard deviation of errors. h. Construct a \(95 \%\) confidence interval for \(B\). i. Test at a \(5 \%\) significance level if \(B\) is positive. j. Using \(\alpha=.025\), can you conclude that the linear correlation coefficient is positive?
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