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Chapter 9: Problem 14

It is not obvious from the formulas, but the values of the sample test statistic \(t\) for the correlation coefficient and for the slope of the least- squares line are equal for the same data set. This fact is based on the relation $$ b=r \frac{s_{y}}{s_{x}} $$ where \(s_{y}\) and \(s_{x}\) are the sample standard deviations of the \(x\) and \(y\) values, respectively. (a) Many computer software packages give the \(t\) value and corresponding \(P\) -value for \(b\). If \(\beta\) is significant, is \(\rho\) significant? (b) When doing statistical tests "by hand," it is easier to compute the sample test statistic \(t\) for the sample correlation coefficient \(r\) than it is to compute the sample test statistic \(t\) for the slope \(b\) of the sample least- squares line. Compare the results of parts (b) and (f) for Problems \(7-12\) of this problem set. Is the sample test statistic \(t\) for \(r\) the same as the corresponding test statistic for \(b\) ? If you conclude that \(\rho\) is positive, can you conclude that \(\beta\) is positive at the same level of significance? If you conclude that \(\rho\) is not significant, is \(\beta\) also not significant at the same level of significance?

Short Answer

Expert verified
Yes, if \( \beta \) is significant, \( \rho \) is significant. \( t \) for \( r \) equals \( t \) for \( b \) for the same data set.

Step by step solution

01

Understanding the Relationship

The exercise revolves around understanding the relationship between the correlation coefficient \( r \) and the slope \( b \) of the least squares line. The formula given is \( b = r \frac{s_y}{s_x} \), implying that both \( r \) and \( b \) are linked through the standard deviations of \( x \) and \( y \). This fact makes their test statistics \( t \) equal for the same data set.
02

Role of Significant \( \beta \) and \( \rho \)

When \( \beta \) (the population slope) is significant, \( \rho \) (population correlation) must also be significant. This is because the significance of \( \beta \) indicates a non-zero slope in the least-squares line, which also means a significant relationship between \( x \) and \( y \), thus making \( \rho \) significant.
03

Comparing Test Statistics

Computing the test statistic \( t \) for \( r \) (correlation coefficient) is typically easier when done manually. The test statistic for \( r \) (given data set conditions) will be equal to the test statistic for \( b \) due to their mathematical relationship, confirming that if one is significant, the other is as well, and both have the same value.
04

Conclusion

If \( \rho \) is positive (or significant), \( \beta \) will also be positive (or significant) at the same level of significance due to the direct dependency described by the formula \( b = r \frac{s_y}{s_x} \). Conversely, if \( \rho \) is not significant, \( \beta \) will not be significant either at the same level of significance.

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Key Concepts

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

Sample Test Statistic
The sample test statistic is an essential part of statistical analysis, especially when determining the relationship between two variables. When we talk about the test statistic for the sample correlation coefficient, represented as \( t \), we're essentially looking at how extreme our sample correlation is, assuming there is no actual correlation in the population (the null hypothesis). The focus in this context is understanding whether our observed correlation \( r \) could easily occur by chance. The calculation of this test statistic for the sample correlation follows a formula that accounts for the sample size and the correlation coefficient:
  • For correlation coefficient \( r \), the formula is \( t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \)
This formula shows that with more data points (or a larger sample size), even a small correlation can become statistically significant. For the slope \( b \) of the least squares line, the test statistic \( t \) is calculated differently, usually making it more cumbersome to compute manually. Nevertheless, for the same data set, both \( t \) values for \( r \) and \( b \) turn out to be the same due to the inherent relationship in the calculations.
Least Squares Line
The least squares line is a fundamental concept in regression analysis, providing a way to summarize the relationship between two quantitative variables. This line represents the line that best fits the data, minimizing the sum of the squared differences between observed and predicted values. Its slope, \( b \), indicates the rate of change in the dependent variable \( y \) for each unit change in the independent variable \( x \).To compute the slope \( b \), we use the relation:\[ b = r \frac{s_y}{s_x}\]Where:
  • \( r \) is the correlation coefficient providing a standardized measure of the strength and direction of the linear relationship between \( x \) and \( y \).
  • \( s_y \) and \( s_x \) are the sample standard deviations of the \( y \) and \( x \) variables, respectively.
This relationship illustrates how the correlation adjusts the measure of how much \( y \) changes per change in \( x \) based on their variabilities.A key takeaway is that a significant slope in the least squares line implies a significant linear relationship, making both \( \beta \) (slope) and \( \rho \) (correlation) simultaneously meaningful in a statistically significant analysis.
Statistical Significance
Statistical significance is a critical concept that allows us to decide whether the observed effects, such as the relationship between two variables, are meaningful or likely just due to random chance. In correlation and regression analysis, determining whether the slope \( \beta \) or correlation coefficient \( \rho \) is significant directly influences how we interpret the data.When a statistic, like our correlation or slope, is considered significant, we conclude that there is enough evidence to suggest that the relationship observed in our sample reflects a real relationship in the population. We use the sample test statistic \( t \) to make this determination:
  • If \( t \) exceeds the critical value from the \( t \)-distribution (which considers our chosen level of significance), we reject the null hypothesis of no relationship.
It's crucial to note:
  • If \( \beta \) is significant, then \( \rho \) is also significant due to their computational relationship.
  • The reverse is also true鈥攊f \( \rho \) isn鈥檛 significant, \( \beta \) typically won't be either.
This interconnected significance ensures consistency across our interpretations, whether focusing on correlation or regression analysis.

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Most popular questions from this chapter

In baseball, is there a linear correlation between batting average and home run percentage? Let \(x\) represent the batting average of a professional baseball player, and let \(y\) represent the player's home run percentage (number of home runs per 100 times at bat). A random sample of \(n=7\) professional baseball players gave the following information (Reference: The Baseball Encyclopedia, Macmillan Publishing Company). $$ \begin{array}{l|lllllll} \hline x & 0.243 & 0.259 & 0.286 & 0.263 & 0.268 & 0.339 & 0.299 \\ \hline y & 1.4 & 3.6 & 5.5 & 3.8 & 3.5 & 7.3 & 5.0 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or high? positive or negative? (c) Use a calculator to verify that \(\Sigma x=1.957, \Sigma x^{2} \approx 0.553, \Sigma y=30.1\), \(\Sigma y^{2}=150.15\), and \(\Sigma x y \approx 8.753 .\) Compute \(r .\) As \(x\) increases, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.

The following data are based on information from Domestic Affairs. Let \(x\) be the average number of employees in a group health insurance plan, and let \(y\) be the average administrative cost as a percentage of claims. $$ \begin{array}{l|rrrrr} \hline x & 3 & 7 & 15 & 35 & 75 \\ \hline y & 40 & 35 & 30 & 25 & 18 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that \(\Sigma x=135, \Sigma x^{2}=7133, \quad \Sigma y=148\), \(\Sigma y^{2}=4674\), and \(\Sigma x y=3040\). Compute \(r\). As \(x\) increases from 3 to 75 , does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.

Let \(x=\) day of observation and \(y=\) number of locusts per square meter during a locust infestation in a region of North Africa. $$ \begin{array}{l|llrrr} \hline x & 2 & 3 & 5 & 8 & 10 \\ \hline y & 2 & 3 & 12 & 125 & 630 \\ \hline \end{array} $$ (a) Draw a scatter diagram of the \((x, y)\) data pairs. Do you think a straight line will be a good fit to these data? Do the \(y\) values almost seem to explode as time goes on? (b) Now consider a transformation \(y^{\prime}=\log y .\) We are using common logarithms of base \(10 .\) Draw a scatter diagram of the \(\left(x, y^{\prime}\right)\) data pairs and compare this diagram with the diagram of part (a). Which graph appears to better fit a straight line? (c) Use a calculator with regression keys to find the linear regression equation for the data pairs \(\left(x, y^{\prime}\right) .\) What is the correlation coefficient? (d) The exponential growth model is \(y=\alpha \beta^{x}\). Estimate \(\alpha\) and \(\beta\) and write the exponential growth equation. Hint: See Problem 22 .

When we use a least-squares line to predict \(y\) values for \(x\) values beyond the range of \(x\) values found in the data, are we extrapolating or interpolating? Are there any concerns about such predictions?

Describe the relationship between two variables when the correlation coefficient \(r\) is (a) near \(-1\). (b) near 0. (c) near 1 .
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