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Chapter 5: Problem 66

The accompanying data represent \(x=\) the amount of catalyst added to accelerate a chemical reaction and \(y=\) the resulting reaction time: $$ \begin{array}{rrrrrr} x & 1 & 2 & 3 & 4 & 5 \\ y & 49 & 46 & 41 & 34 & 25 \end{array} $$ a. Calculate \(r\). Does the value of \(r\) suggest a strong linear relationship? b. Construct a scatterplot. From the plot, does the word linear really provide the most effective description of the relationship between \(x\) and \(y\) ? Explain.

Short Answer

Expert verified
a. The calculated correlation coefficient \(r = -0.98\) suggests a strong negative linear relationship between the amount of catalyst and the reaction time. b. The scatter plot suggests that the word 'linear' aptly describes the relationship between the two variables, indicating that as the amount of catalyst increases, the reaction time decreases significantly.

Step by step solution

01

Calculation of correlation coefficient (r)

Correlation coefficient (r) between two variables can be calculated using the formula \(r = \frac{{n(\sum xy) - (\sum x)(\sum y)}}{{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}}\). By looking at the given data: for \(x\), \(\sum x = 15\), \(\sum x^2 = 55\), for \(y\), \(\sum y = 195\) and \(\sum y^2 = 8951\), and for \(xy\), \(\sum xy = 480\). Plugging these values to the formula gives \(r = -0.98\). The magnitude of this value is close to 1 which suggests a strong linear relationship but the negative sign indicates an inverse relationship; as \(x\) increases, \(y\) decreases.
02

Drawing scatter plot

To draw the scatter plot, plot the \(x\) values on the x-axis and corresponding \(y\) values on the y-axis. Five points will be in the scatter plot representing each pairing of (\(x\), \(y\)). The plotted points appear to follow a decreasing linear trend, supporting that there is a strong negative linear relationship between \(x\) and \(y\).
03

Interpretation

The word 'linear' adequately describes the relationship between \(x\) and \(y\) as both the correlation coefficient and the scatter plot suggest a strong linear relationship. However, it is important to mention that the relationship is negative or decreasing; as the amount of catalyst increases, the reaction time decreases, implying the negative correlation.

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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 alue of \(y\) for which \(r>.5\). c. If a fifth observation is made at the value \(x=6\), find 芒 value of \(y\) for which \(r<.5\).

Explain why the slope \(b\) of the least-squares line always has the same sign (positive or negative) as does the sample correlation coefficient \(r\).

The relationship between hospital patient-to-nurse ratio and various characteristics of job satisfaction and patient care has been the focus of a number of research studies. Suppose \(x=\) patient-to-nurse ratio is the predictor variable. For each of the following potential dependent variables, indicate whether you expect the slope of the least-squares line to be positive or negative and give a brief explanation for your choice. a. \(y=\) a measure of nurse's job satisfaction (higher values indicate higher satisfaction) b. \(y=\) a measure of patient satisfaction with hospital care (higher values indicate higher satisfaction) c. \(y=\) a measure of patient quality of care.

The sample correlation coefficient between annual raises and teaching evaluations for a sample of \(n=353\) college faculty was found to be \(r=.11\) ("Determination of Faculty Pay: An Agency Theory Perspective," Academy of Management Joumal [1992]: 921-955). a. Interpret this value. b. If a straight line were fit to the data using least squares, what proportion of variation in raises could be attributed to the approximate linear relationship between raises and evaluations?

The paper "Increased Vital and Total Lung Capacities in Tibetan Compared to Han Residents of Lhasa" \((\) American Journal of Physical Anthropology [1991]: \(341-351\) ) included a scatterplot of vital capacity \((y)\) versus chest circumference \((x)\) for a sample of 16 Tibetan natives, from which the following data were read: \(\begin{array}{rrrrrrrrr}x & 79.4 & 81.8 & 81.8 & 82.3 & 83.7 & 84.3 & 84.3 & 85.2 \\ y & 4.3 & 4.6 & 4.8 & 4.7 & 5.0 & 4.9 & 4.4 & 5.0 \\ x & 87.0 & 87.3 & 87.7 & 88.1 & 88.1 & 88.6 & 89.0 & 89.5 \\ .1 & 6.1 & 4.7 & 57 & 5.7 & 5.2 & 5.5 & 5.0 & 5.3\end{array}\) \(87.3\) \(4.7\) \(87.7\) \(5.7\) \(\begin{array}{rrrr}1 & 88.1 & 88.6 & 89.0 \\ 7 & 5.2 & 5.5 & 5.0\end{array}\) \(x\) \(y\) \(6.1\) \(8.1\) \(5.7\) a. Construct a scatterplot. What does it suggest about the nature of the relationship between \(x\) and \(y\) ? b. The summary quantities are $$ \sum x=1368.1 \quad \sum y=80.9 $$ \(\sum x y=6933.48 \quad \sum x^{2}=117,123.85 \quad \sum y^{2}=412.81\) Verify that the equation of the least-squares line is \(\hat{y}=\) \(-4.54+0.1123 x\), and draw this line on your scatterplot. c. On average, roughly what change in vital capacity is associated with a \(1-\mathrm{cm}\) increase in chest circumference? with a \(10-\mathrm{cm}\) increase? d. What vital capacity would you predict for a Tibetan native whose chest circumference is \(85 \mathrm{~cm}\) ? e. Is vital capacity completely determined by chest circumference? Explain.
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