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Chapter 5: Problem 2

Is the following statement correct? Explain why or why not. A correlation coefficient of 0 implies that no relationship exists between the two variables under study.

Short Answer

Expert verified
The statement is partially correct. A correlation coefficient of 0 does mean there's no linear relationship between the two variables. However, it doesn鈥檛 entirely rule out the possibility of other types of relationships; for instance, non-linear relationships are not captured by the correlation coefficient.

Step by step solution

01

Understanding the Correlation Coefficient

Start analyzing the statement with an understanding of the correlation coefficient. The correlation coefficient is a statistical measure that helps us understand whether and how strongly pairs of variables are related.
02

Implications of a Correlation coefficient of 0

A correlation coefficient of 0 signifies there is no linear relationship between the two variables being studied. This means changes in the value of one variable do not cause a consistent and proportionate change in the value of the other variable in a predictable linear way.
03

Exploring Potential Non-linear Relationships

Although a correlation coefficient of 0 implies no linear relationship between the variables, it's important to note that there might be other forms of relationship between the variables. For instance, the relationship could be quadratic, exponential or any other non-linear form that the correlation coefficient is not designed to capture.

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Most popular questions from this chapter

Consider the four \((x, y)\) pairs \((0,0),(1,1)\), \((1,-1)\), and \((2,0)\) a. What is the value of the sample correlation coefficient \(r\) ? b. If a fifth observation is made at the value \(x=6\), find a value of \(y\) for which \(r>0.5\). c. If a fifth observation is made at the value \(x=6\), find a value of \(y\) for which \(r<0.5\).

It may seem odd, but one of the ways biologists can tell how old a lobster is involves measuring the concentration of a pigment called neurolipofuscin in the eyestalk of a lobster. (We are not making this up!) The authors of the paper "Neurolipofusdin is a Measure of Age in Panulirus argus, the Caribbean Spiny Lobster. in Florida" (Biological Bulletin [2007]: \(55-66\) ) wondered if it was sufficient to measure the pigment in just one eye stalk, which would be the case if there is a strong relationship between the concentration in the right and left eyestalks. Pigment concentration (as a percentage of tissue sample) was measured in both eyestalks for 39 lobsters, resulting is the following summary quantities (based on data read from a graph that appeared in the paper): $$ \begin{aligned} n &=39 & \sum x &=88.8 & & \Sigma y &=86.1 \\ \Sigma x y &=281.1 & \sum x^{2} &=288.0 & & \Sigma y^{2}=286.6 \end{aligned} $$ An alternative formula for computing the correlation coefficient that is based on raw data and is algebraically equivalent to the one given in the text is $$ r=\frac{\sum x y-\frac{\left(\sum x\right)(\Sigma y)}{n}}{\sqrt{\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}} \sqrt{\sum y^{2}-\frac{\left(\sum y\right)^{2}}{n}}} $$ Use this formula to compute the value of the correlation coefficient, and interpret this value.

The paper "A Cross-National Relationship Between Sugar Consumption and Major Depression?" (Depression and Anxiety [2002]: \(118-120\) ) concluded that there was a correlation between refined sugar consumption (calories per person per day) and annual rate of major depression (cases per 100 people) based on data from six countries. The following data were read from a graph that appeared in the paper: \begin{tabular}{lcc} & Sugar & Depression \\ Country & Consumption & Rate \\ \hline Korea & 150 & \(2.3\) \\ United States & 300 & \(3.0\) \\ France & 350 & \(4.4\) \\ Germany & 375 & \(5.0\) \\ Canada & 390 & \(5.2\) \\ New Zealand & 480 & \(5.7\) \\ \hline \end{tabular} a. Compute and interpret the correlation coefficient for this data set. b. Is it reasonable to conclude that increasing sugar consumption leads to higher rates of depression? Explain. c. Do you have any concerns about this study that would make you hesitant to generalize these conclusions to other countries?

The paper "Developmental and Individual Differences in Pure Numerical Estimation" (Developmental Psychology [2006]: \(189-201)\) describes a study of how young children develop the ability to estimate lengths. Children were shown a piece of paper with two lines. One line was a short line labeled as having length zip. The second line was a much longer line labeled as having length 1000 zips. The child was then asked to draw a line that had a length of a specified number of zips, such as 438 zips. The data in the accompanying table gives the length requested and the average of the actual lengths of the lines drawn by 30 second graders. \begin{tabular}{cc} Requested Length & Second Grade Average Length Drawn \\ \hline 3 & \(37.15\) \\ 7 & \(92.88\) \\ 19 & \(207.43\) \\ 52 & \(272.45\) \\ 103 & \(458.20\) \\ 158 & \(442.72\) \\ 240 & \(371.52\) \\ 297 & \(467.49\) \\ 346 & \(487.62\) \\ 391 & \(530.96\) \\ 438 & \(482.97\) \\ 475 & \(544.89\) \\ 502 & \(515.48\) \\ 586 & \(595.98\) \\ 613 & \(575.85\) \\ 690 & \(605.26\) \\ 721 & \(637.77\) \\ 760 & \(674.92\) \\ 835 & \(701.24\) \\ 874 & \(662.54\) \\ 907 & \(758.51\) \\ 962 & \(749.23\) \\ \hline \end{tabular} a. Construct a scatterplot of \(y=\) second grade average length drawn versus \(x=\) requested length. b. Based on the scatterplot in Part (a), would you suggest using a line, a quadratic curve, or a cubic curve to describe the relationship between \(x\) and \(y\) ? Explain choice c. Using a statistical software package or a graphing calculator, fit a cubic curve to this data and use it to predict average length drawn for a requested length of 500 zips.

The following data on sale price, size, and land-to-building ratio for 10 large industrial properties appeared in the paper "Using Multiple Regression Analysis in Real Estate Appraisal" (Appraisal Journal \([2002]: 424-430):\) \begin{tabular}{rrrr} & Sale Price (millions of dollars) & Size (thousands of sq. ft.) & Land- toBuilding \\ \hline 1 & \(10.6\) & 2166 & \(2.0\) \\ 2 & \(2.6\) & 751 & \(3.5\) \\ 3 & \(30.5\) & 2422 & \(3.6\) \\ 4 & \(1.8\) & 224 & \(4.7\) \\ 5 & \(20.0\) & 3917 & \(1.7\) \\ 6 & \(8.0\) & 2866 & \(2.3\) \\ 7 & \(10.0\) & 1698 & \(3.1\) \\ 8 & \(6.7\) & 1046 & \(4.8\) \\ 9 & \(5.8\) & 1108 & \(7.6\) \\ 10 & \(4.5\) & 405 & \(17.2\) \\ \hline \end{tabular} a. Calculate and interpret the value of the correlation coefficient between sale price and size. b. Calculate and interpret the value of the correlation coefficient between sale price and land-to-building ratio. c. If you wanted to predict sale price and you could use either size or land- to-building ratio as the basis for making predictions, which would you use? Explain. d. Based on your choice in Part (c), find the equation of the least-squares regression line you would use for predicting \(y=\) sale price.
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