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Chapter 13: Problem 54

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

Short Answer

Expert verified
The linear correlation coefficient is obtained by substituting the given values into the Pearson's r formula and solving the resulting mathematical equation.

Step by step solution

01

Substitute Given Values

Substitute the provided summation values into the formula to find the correlation coefficient. Let's replace \(N = 250\), \(\sum x = 9880\), \(\sum y = 1456\), \(\sum xy = 85080\), \(\sum x^2 = 485870\), and \(\sum y^2 = 135675\).
02

Compute Product of Bilateral Summations

To simplify the calculation, first find the product of \(\sum x\) and \(\sum y\), the product of \(\sum x\) twice and the product of \(\sum y\) twice.
03

Compute Difference in Squares

Now compute the differences \(N\sum x^2 - (\sum x)^2\) and \(N\sum y^2 - (\sum y)^2\). This will simplify the denominator for finding the coefficient.
04

Compute the Numerator and the Denominator

compute the difference \(N\sum xy - \sum x \sum y\) and the square root of the product from step 3. This will provide the numerator and the denominator for the fraction.
05

Find the Linear Correlation Coefficient

The linear correlation coefficient \(r\) is the result of the division of the numerator by the denominator from step 4.

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Key Concepts

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

Population Data Set
In statistical analysis, a population data set encompasses all possible observations relevant to a particular experiment. Think of it as the complete collection of data points. Here, the data set includes several summation values that are crucial for calculating various statistical measures, including the linear correlation coefficient. The population size, denoted by \(N\), plays an essential role in computations as it controls the scope and impact of the observations. For example:
  • \(N = 250\): This indicates the total number of observations in our study.
  • Understanding your population helps in interpreting the statistical results correctly.
It is essential to differentiate between sample and population data, as calculations may vary based on this distinction.
Summation Values
Summation values are aggregated totals that summarize specific properties of the data set. These values serve as the building blocks for more intricate statistical computations. Key summation values in this context are:
  • \(\Sigma x = 9880\)
  • \(\Sigma y = 1456\)
  • \(\Sigma xy = 85,080\)
  • \(\Sigma x^2 = 485,870\)
  • \(\Sigma y^2 = 135,675\)
These totals make up the basic components of calculating the linear correlation coefficient. By using these summation values, we can better understand the relationships and patterns present in the data. They help streamline calculations by reducing the complexity associated with dealing with individual observations.
Numerator and Denominator
In statistics, the numerator and denominator in a formula can significantly affect the outcome, especially when calculating the linear correlation coefficient (\(r\)). Here:
  • **Numerator:** \(N\sum xy - (\sum x)(\sum y)\), which measures the degree of covariance between the two variables.
  • **Denominator:** The square root of the product of \(N\sum x^2 - (\sum x)^2\) and \(N\sum y^2 - (\sum y)^2\). This represents the combined variance of the variables.
Computing these expressions involves substituting the summation values. The linear correlation coefficient \(r\) is then derived by dividing the numerator by this denominator. This ratio gives insight into the strength and direction of the linear relationship between two variables.
Difference in Squares
The concept of 'difference in squares' refers to subtracting the square of a summation from a multiple of another summation. Within the context of calculating the correlation coefficient, we compute differences like:
  • \(N\sum x^2 - (\sum x)^2\)
  • \(N\sum y^2 - (\sum y)^2\)
These differences are crucial for the denominator in the correlation formula. They represent the squared differences of the variables. Why is this important? It adjusts for the influence of each variable's sum, removing biases due to scale differences. Hence, these calculations ensure that the correlation coefficient accurately reflects the relationship, accounting for variability in the data.

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Most popular questions from this chapter

The following table gives information on the calorie count and grams of fat for 8 of the many types of bagels produced and sold by Panera Bread. $$ \begin{array}{lcc} \hline \text { Bagel } & \text { Calories } & \text { Fat (grams) } \\ \hline \text { Asiago Cheese } & 330 & 6.0 \\ \text { Blueberry } & 340 & 1.5 \\ \text { Cinnamon Crunch } & 420 & 6.0 \\ \text { Cinnamon Swirl \& Raisin } & 320 & 2.0 \\ \text { Everything } & 300 & 2.5 \\ \text { French Toast } & 350 & 4.0 \\ \text { Plain } & 290 & 1.5 \\ \text { Sesame } & 310 & 3.0 \\ \hline \end{array} $$ With calories as the dependent variable and fat content as the independent variable, find the following: a. \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Standard deviation of errors c. SST, SSE, and SSR d. Coefficient of determination

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 values of \(\sigma_{e}\) and \(\rho^{2}\).

Construct a \(99 \%\) confidence interval for the mean value of \(y\) and a \(99 \%\) prediction interval for the predicted value of \(y\) for the following. a. \(\hat{y}=3.25+.80 x\) for \(x=15\) given \(s_{e}=.954, \bar{x}=18.52, \mathrm{SS}_{x x}=\) \(144.65\), and \(n=10\) b. \(\hat{y}=-27+7.67 x\) for \(x=12\) given \(s_{e}=2.46, \bar{x}=13.43, \mathrm{SS}_{x x}=\) \(369.77\), and \(n=10\)

For a sample data set, the slope \(b\) of the regression line has a negative value. Which of the following is true about the linear correlation coefficient \(r\) calculated for the same sample data? a. The value of \(r\) will be positive. b. The value of \(r\) will be negative. c. The value of \(r\) can be positive or negative.

For a sample data set, the linear correlation coefficient \(r\) has a positive value. Which of the following is true about the slope \(b\) of the regression line estimated for the same sample data? a. The value of \(b\) will be positive. b. The value of \(b\) will be negative. c. The value of \(b\) can be positive or negative.
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