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Chapter 13: Problem 68

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 \(\rho\).

Short Answer

Expert verified
The evaluation of \( \rho \) will provide the short answer.

Step by step solution

01

Identify given variables

First, identify the variables needed to calculate the Pearson correlation coefficient \( \rho \) from the given data. Here, \( N=250 \), \( \Sigma x=9880 \), \( \Sigma y=1456 \), \( \Sigma x y=85080 \), \( \Sigma x^{2}=485870 \) and \( \Sigma y^{2}=135675 \)
02

Apply formula for Pearson correlation coefficient

Then, apply the formula for the calculation of the Pearson correlation coefficient ( \( \rho \)):\[ \rho=\frac{N(\Sigma x y)-(\Sigma x)(\Sigma y)}{\sqrt{[N(\Sigma x^{2})-(\Sigma x)^{2}][N(\Sigma y^{2})-(\Sigma y)^{2}]}}\]
03

Substitute variables into formula

Substitute the known variables into the formula:\[\rho=\frac{250*85080-9880*1456}{\sqrt{(250*485870-9880^{2})*(250*135675-1456^{2})}}\]
04

Compute the correlation coefficient

Finally, calculate all terms separately before simplifying the fraction.\n In the end, you will obtain the value of \( \rho \).

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Key Concepts

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

Understanding Linear Correlation
Linear correlation is a statistical measure that describes the strength and direction of the relationship between two variables. When two variables are linearly correlated, it means that changes in one variable are consistently associated with changes in another variable in a specific pattern.
This pattern can often be represented by a straight line when graphed, hence the term "linear."
  • If the correlation is positive, as one variable increases, the other tends to increase as well.
  • If the correlation is negative, as one variable increases, the other tends to decrease.
  • The strength of the correlation is measured on a scale from -1 to 1.
A correlation of 1 implies a perfect positive linear correlation, -1 implies a perfect negative linear correlation, and 0 implies no correlation. To understand how two variables interact, the correlation coefficient \( \rho \) is calculated using the given data.
Exploring a Population Data Set
In statistics, a population data set refers to the complete set of data collected from every member of a population of interest. This can encompass a broad array of subjects, such as people, objects, or events, and contains all potential data points.
The uniqueness of working with a full population data set is that it can provide comprehensive insights into population metrics without the need to infer from a sample.
Knowing the total population size, denoted as \( N \), along with the sum of all observations, helps in calculating important statistical measures.
  • \( \Sigma x \) and \( \Sigma y \) represent the sum of all x and y values, respectively.
  • These sums allow us to compute other values, such as means, variances, and covariances.
  • Accurate computation helps in deriving correlation measures like the Pearson correlation coefficient.
Overall, analyzing a population data set gives a complete statistical picture of the data in discussion.
Basics of Statistics
Statistics is the study of collecting, analyzing, interpreting, presenting, and organizing data. It forms the foundation for understanding and working with data. One of the primary goals of statistics is to determine the relationship between variables and to make inferences about populations based on data from samples.
Various statistical methods help in uncovering patterns, testing hypotheses, and drawing conclusions from data.
  • Descriptive statistics summarize data features, often via measures such as mean, median, or standard deviation.
  • Inferential statistics enable predictions and inferences about a larger dataset through sample data analysis.
  • The correlation coefficient is among these statistical tools, offering insights about linear relationships between variables.
Statistics provide invaluable insights into various fields, helping to inform decisions and validate scientific and observational studies.
Covariance: A Measure of Joint Variability
Covariance is a statistical measure that indicates the extent to which two variables change together. It is pivotal for understanding how two variables co-vary from their respective means, which forms the basis for calculating correlation.
If two variables tend to increase together, the covariance is positive. Conversely, if one increases when the other decreases, the covariance is negative.
  • Covariance by itself doesn't provide a standardized measure of variable relationships since its value depends on the units of the variables.
  • To get a standardized view, it must be divided by their standard deviations, leading to the correlation coefficient.
  • In the formula for the linear correlation coefficient, covariance represents the numerator part: \( N(\Sigma x y) - (\Sigma x)(\Sigma y) \).
Understanding covariance helps to grasp how variables are linked and is essential before moving to the more refined measure of correlation.

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Most popular questions from this chapter

For a sample data set on two variables, the value of the linear correlation coefficient is zero. Does this mean that these variables are not related? Explain

Plot the following straight lines. Give the values of the \(y\) -intercept and slope for each of these lines and interpret them. Indicate whether each of the lines gives a positive or a negative relationship between \(x\) and \(y\) a. \(y=100+5 x \quad\) b. \(y=400-4 x\)

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 2009. $$ \begin{array}{l|llllllllllll} \hline \text { Warranty (thousands of miles) } & 60 & 70 & 75 & 50 & 80 & 55 & 65 & 65 & 70 & 65 & 60 & 65 \\ \hline \text { Price per tire }(\$) & 95 & 70 & 94 & 90 & 121 & 70 & 84 & 80 & 92 & 79 & 66 & 95 \\ \hline \end{array} $$ 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 \(\mathrm{SS}_{x y}\) 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 \(\mathrm{b}\). 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 the \(5 \%\) significance level if \(B\) is positive. j. Using \(\alpha=.025\), can you conclude that the linear correlation coefficient is positive?

The following table gives the completion times for the winners in the women's 200 -meter dash finals in the Summer Olympic Games from 1972 to 2008 . The times are in seconds rounded to the nearest \(1 / 100\) second. $$ \begin{array}{cc} \hline \text { Olympic Year } & \text { Time (seconds) } \\ \hline 1972 & 22.40 \\ 1976 & 22.37 \\ 1980 & 22.03 \\ 1984 & 21.81 \\ 1988 & 21.34 \\ 1992 & 21.81 \\ 1996 & 22.12 \\ 2000 & 21.85 \\ 2004 & 22.05 \\ 2008 & 21.74 \\ \hline \end{array} $$ a. Assign a value of 0 to 1972,1 to 1976,2 to 1980 , and so on. Call this new variable Year. Make a new table with the variables Year and Time. b. With year as an independent variable and time as the dependent variable, compute \(S S_{x x}, S S_{y y}\) and \(S S_{x x}\) construct a scatter diagram for these data. Does the scatter diagram exhibit a linear negative relationship between year and time? d. Find the least squares regression line \(\hat{y}=a+b x\). e. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part \(\mathrm{d}\). f. Compute the correlation coefficient \(r\). g. Predict the time for the year 2016 . Comment on this prediction.

The following table containing data on the aerobic exercise levels (running distance in miles) and blood sugar levels for 12 different days for a diabetic is reproduced from Exercise \(13.27 .\) $$ \begin{array}{l|rrrrrrrrrrr} \hline \text { Distance (miles) } & 2 & 2 & 2.5 & 2.5 & 3 & 3 & 3.5 & 3.5 & 4 & 4 & 4.5 & 4.5 \\ \hline \text { Blood sugar }(\mathrm{mg} / \mathrm{dL}) & 136 & 146 & 131 & 125 & 120 & 116 & 104 & 95 & 85 & 94 & 83 & 75 \\ \hline \end{array} $$ a. Find the standard deviation of errors. b. Compute the coefficient of determination. What percentage of the variation in blood sugar level is explained by the least squares regression of blood sugar level on the distance run? What percentage of this variation is not explained?
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