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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
The Pearson's correlation coefficient \(r\) is -0.9685, which indicates a strong negative linear relationship between the amount of catalyst and the reaction time. The scatterplot confirms this strong negative linear dependence between the two variables.

Step by step solution

01

Calculation of \(\overline{x}\), \(\overline{y}\), \(s_x\), and \(s_y\)

For \(x = \{1, 2, 3, 4, 5\}\): \(\overline{x} = \frac{1+2+3+4+5}{5} = 3, s_x = \sqrt{\frac{(1-3)^2+(2-3)^2+(3-3)^2+(4-3)^2+(5-3)^2}{5-1}} = \sqrt{2.5}\)For \(y = \{49, 46, 41, 34, 25\}\): \(\overline{y} = \frac{49+46+41+34+25}{5} = 39, s_y = \sqrt{\frac{(49-39)^2+(46-39)^2+(41-39)^2+(34-39)^2+(25-39)^2}{5-1}} = \sqrt{74.5}\)
02

Calculation of \(r\)

Using the formula \(r = \frac{\Sigma(xy) - n\overline{x}\overline{y}}{(n-1)s_xs_y}\) where n is the number of observations and \(\Sigma(xy)\) is the sum of the products of corresponding values, we get \(r = \frac{(1*49+2*46+3*41+4*34+5*25) - 5*3*39}{(5-1)\sqrt{2.5}\sqrt{74.5}} = -0.9685\) Hence, the value suggests a strong negative linear relationship between \(x\) and \(y\).
03

Constructing the scatterplot

When plotting the data points following the \(x\) and \(y\) coordinates and connecting the dots, a descending line from left to right is formed indicating a strong negative correlation.
04

Scatterplot Interpretation

Looking at the scatterplot, the negative linear relationship seems to be an effective description of the relationship between \(x\) and \(y\), because a line can easily model the descending data points (as \(x\) increases, \(y\) decreases). This description matches with the negative correlation described by \(r\) earlier.

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Key Concepts

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

Pearson Correlation Coefficient
When measuring the strength and direction of a linear relationship between two continuous variables, the Pearson correlation coefficient, symbolized by r, is a statistic that plays a pivotal role. The value of r ranges from -1 to 1, where -1 indicates a perfect negative linear correlation, 1 signifies a perfect positive linear correlation, and 0 denotes no linear relationship at all.

In the context of the exercise provided, the calculation of r involves finding the mean of the x and y values, their respective standard deviations, and the sum of the product of corresponding values. The strong negative value of r = -0.9685 here suggests that as the amount of catalyst (x) increases, the reaction time (y) decreases in a consistent pattern, indicating a significantly inverse linear relationship between these two variables.
Scatterplot
A scatterplot is a type of graph used to visually display the relationship between two numeric variables. Each point on the scatterplot corresponds to a pair of x and y values from our dataset. By plotting the amount of catalyst against the reaction time from the textbook exercise, we can visualize how the variables correlate with each other.

Upon constructing the scatterplot for our exercise, we can observe how the data points create a downward trend as we move from left to right on the graph, which illustrates the strong negative correlation computed previously with the Pearson correlation coefficient. This visual representation reinforces the calculated value of r, making the concept of linear relationship more tangible.
Standard Deviation
The standard deviation is a measure that reflects the amount of variation or dispersion from the average. In simpler terms, it tells us how spread out the numbers in a data set are. A low standard deviation means that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wide range.

For both x and y in the exercise, we calculate the standard deviation (\(s_x\text{ and }s_y\text{, respectively}\)) as part of determining the Pearson correlation coefficient. The standard deviations are crucial for understanding the variability within our data, which in turn helps to inform the significance of the correlation coefficient. For example, if the standard deviation of y was very low, small changes in x could have a significant impact on y, contributing to the strong correlation observed.

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Most popular questions from this chapter

Data on high school GPA \((x)\) and first-year college GPA ( \(y\) ) collected from a southeastern public research university can be summarized as follows ("First-Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students," Journal of College Student Development \([1999]: 599-605):\) $$ \begin{array}{clc} n=2600 & \sum x=9620 & \sum y=7436 \\ \sum x y=27,918 & \sum x^{2}=36,168 & \sum y^{2}=23,145 \end{array} $$ a. Find the equation of the least-squares regression line. b. Interpret the value of \(b\), the slope of the least-squares line, in the context of this problem. c. What first-year GPA would you predict for a student with a \(4.0\) high school GPA?

The paper "Biomechanical Characteristics of the Final Approach Step, Hurdle, and Take-Off of Elite American Springboard Divers" (Journal of Human Movement Studies [1984]: 189-212) gave the following data on \(y=\) judge's score and \(x=\) length of final step (in meters) for a sample of seven divers performing a forward pike with a single somersault: $$ \begin{array}{cccccccc} y & 7.40 & 9.10 & 7.20 & 7.00 & 7.30 & 7.30 & 7.90 \\ x & 1.17 & 1.17 & 0.93 & 0.89 & 0.68 & 0.74 & 0.95 \end{array} $$ a. Construct a scatterplot. b. Calculate the slope and intercept of the least-squares line. Draw this line on your scatterplot. c. Calculate and interpret the value of Pearson's sample correlation coefficient.

For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to \(0 .\) Explain your choice. a. Maximum daily temperature and cooling costs b. Interest rate and number of loan applications c. Incomes of husbands and wives when both have full- time jobs d. Height and IQ e. Height and shoe size f. Score on the math section of the SAT exam and score on the verbal section of the same test g. Time spent on homework and time spent watching television during the same day by elementary school children h. Amount of fertilizer used per acre and crop yield (Hint: As the amount of fertilizer is increased, yield tends to increase for a while but then tends to start decreasing.)

The following data on degree of exposure to \({ }^{242} \mathrm{Cm}\) alpha particles \((x)\) and the percentage of exposed cells without aberrations \((y)\) appeared in the paper "Chromosome Aberrations Induced in Human Lymphocytes by D-T Neutrons" (Radiation Research \([1984]: 561-573)\) : $$ \begin{array}{rrrrr} x & 0.106 & 0.193 & 0.511 & 0.527 \\ y & 98 & 95 & 87 & 85 \\ x & 1.08 & 1.62 & 1.73 & 2.36 \\ y & 75 & 72 & 64 & 55 \\ x & 2.72 & 3.12 & 3.88 & 4.18 \\ y & 44 & 41 & 37 & 40 \end{array} $$ Summary quantities are $$ \begin{gathered} n=12 \quad \sum x=22.027 \quad \sum y=793 \\ \sum x^{2}=62.600235 \quad \sum x y=1114.5 \quad \sum y^{2}=57,939 \end{gathered} $$ a. Obtain the equation of the least-squares line. b. Construct a residual plot, and comment on any interesting features.

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.
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