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Chapter 13: Problem 70

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?

Short Answer

Expert verified
First, calculate the mean of \( x \) and \( y \). Second, substitute the given values into the formula of \( r \) to compute it. Third, compare the calculated t-value with the critical t-value to determine whether \( \rho \) is significantly different from zero or not. The specific answers depend on the results of the calculations.

Step by step solution

01

Calculate the Mean of X and Y

The mean of \( x \) is given by \( \overline{x} = \Sigma x / n = 100 / 10 = 10 \). Similarly, the mean of \( y \) is \( \overline{y} = \Sigma y / n = 220 / 10 = 22 \).
02

Compute the correlation coefficient \( r \)

The formula for the correlation coefficient is given by \( r = (n \Sigma xy - \Sigma x \Sigma y) / \sqrt{[(n \Sigma x^2 - (\Sigma x)^2)(n \Sigma y^2 - (\Sigma y)^2)]} \). Substituting the given values into the formula, we get \( r = (10 * 3680 - 100 * 220) / \sqrt{[(10 * 1140 - 100 * 100)(10 * 25272 - 220 * 220)]} \). Simplify this expression to calculate the value of \( r \).
03

Determine whether \( \rho \) is different from zero

To test if \( \rho \) is statistically significantly different from zero, one must use a t-test. The t-value is given by \( t = r \sqrt{(n - 2) / (1 - r^2)} \). Find the critical t-value at the 2% significance level. If the absolute value of calculated t-value is greater than the absolute value of the critical t-value, one can then conclude that \( \rho \) is significantly different from zero.

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Most popular questions from this chapter

Briefly explain the assumptions of the population regression model.

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The recommended air pressure in a basketball is between 7 and 9 pounds per square inch (psi). When dropped from a height of 6 feet, a properly inflated basketball should bounce upward between 52 and 56 inches (http://www.bestsoccerbuys.com/balls-basketball.html). The basketball coach at a local high school purchased 10 new basketballs for the upcoming season, inflated the balls to pressures between 7 and \(9 \mathrm{psi}\), and performed the bounce test mentioned above. The data obtained are given in the following table. $$ \begin{array}{l|rrrrrrrrrr} \hline \text { Pressure (psi) } & 7.8 & 8.1 & 8.3 & 7.4 & 8.9 & 7.2 & 8.6 & 7.5 & 8.1 & 8.5 \\ \hline \text { Bounce height (inches) } & 54.1 & 54.3 & 55.2 & 53.3 & 55.4 & 52.2 & 55.7 & 54.6 & 54.8 & 55.3 \\ \hline \end{array} $$ a. With the pressure as an independent variable and bounce height as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Find the least squares regression line. c. Interpret 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. Compute the standard deviation of errors. f. Predict the bounce height of a basketball for \(x=8.0\). g. Construct a \(98 \%\) confidence interval for \(B\). h. Test at a \(5 \%\) significance level whether \(B\) is different from zero. i. Using \(\alpha=.05\), can you conclude that \(\rho\) is different from zero?

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