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Chapter 5: Problem 3

Draw two scatterplots, one for which \(r=1\) and a second for which \(r=-1\)

Short Answer

Expert verified
Two scatterplots have been created. The first one demonstrates a perfect positive correlation (r=1), meaning that the data points align in an ascending straight line. The second one shows a perfect negative correlation (r=-1), where the data points align in a descending straight line.

Step by step solution

01

Preparing the Environment

Firstly, let's set up an environment for creating scatterplots by using a proper programming environment that supports graphical representation, such as Python Matplotlib.
02

Creating Scatterplot for r=1

Next, create a set of x-coordinates and a matching set of y-coordinates, both in an ascending sequence. Since the correlation is positive, as the x values are increasing, the corresponding y values should also increase. Once the data points are created, plot these points using proper graphical representation function.
03

Creating Scatterplot for r=-1

Once again create a set of x-coordinates, but this time pair them up with y-coordinates in a descending sequence. Because the correlation here is negative, as the x values are increasing, the corresponding y values must decrease. Lastly, plot these points.

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Key Concepts

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

Correlation Coefficient
Understanding the correlation coefficient is crucial for interpreting the strength and direction of the relationship between two variables. It's represented by the symbol \( r \), and its value ranges from -1 to 1. A correlation coefficient close to \( r=1 \) signifies a strong positive correlation, meaning as one variable increases, the other tends to increase as well. Conversely, a correlation coefficient of \( r=-1 \) indicates a strong negative correlation, where one variable's increase leads to the other鈥檚 decrease. When \( r \) is around 0, it suggests no linear relationship between the variables.
r=1 Scatterplot
A scatterplot for which \( r=1 \) displays a perfect linear relationship between two variables. In such a scenario, all the data points fall exactly on a straight line with a positive slope. This kind of representation demonstrates that the variables change at the same rate, and the straight line formed indicates a predictable relationship where knowing the value of one variable lets you perfectly predict the value of the other.
r=-1 Scatterplot
Contrary to the \( r=1 \) scatterplot, an \( r=-1 \) scatterplot shows a perfect negative linear correlation. Here, as one variable increases, the other perfectly decreases in response, hence the points fall on a straight line with a negative slope. This graphical representation is powerful because it reveals an inversely proportional relationship between the two variables involved.
Python Matplotlib
Python Matplotlib is an incredibly versatile library for creating static, interactive, and animated visualizations in Python. It's highly favored for crafting scatterplots due to its ease of use and efficiency. To initiate a scatterplot using Python Matplotlib, first one has to import the library, set up a plotting environment, then use specific plotting functions like \( plt.scatter \) for the actual plot creation. With Python Matplotlib, customizing the plots with titles, labels, and different scales is straightforward, which aids in creating a clear graphical representation.
Graphical Representation
A graphical representation plays a fundamental role in data analysis by converting numerical data into visual form. It allows for immediate insights that might not be evident from raw data. Graphical methods, such as scatterplots, help to reveal trends, relationships, and outliers, making complex data more accessible and understandable. Good graphical representation, as facilitated by tools like Python Matplotlib, provides a visual story that complements statistical measures such as the correlation coefficient.

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Most popular questions from this chapter

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.

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 fulltime 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.)

A sample of 548 ethnically diverse students from Massachusetts were followed over a 19 -month period from 1995 and 1997 in a study of the relationship between TV viewing and eating habits ( Pediatrics [2003]: 1321-1326). For each additional hour of television viewed per day, the number of fruit and vegetable servings per day was found to decrease on average by 0.14 serving. a. For this study, what is the dependent variable? What is the predictor variable? b. Would the least-squares line for predicting number of servings of fruits and vegetables using number of hours spent watching TV as a predictor have a positive or negative slope? Explain.

Some plant viruses are spread by insects and tend to spread from the edges of a field inward. The data on \(x=\) distance from the edge of the field (in meters) and \(y=\) proportion of plants with virus symptoms that appeared in the paper "Patterns of Spread of Two NonPersistently Aphid-Borne Viruses in Lupin Stands" \((A n-\) nals of Applied Biology [2005]: \(337-350\) ) was used to fit a least-squares regression line to describe the relationship between \(x\) and \(y^{\prime}=\ln \left(\frac{p}{1-p}\right) .\) Minitab output resulting from fitting the least-squares line is given below. The regression equation is \(\ln (p /(1-p))=-0.917-0.107\) Distance to Crop Edge \(\begin{array}{lrrrr}\text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P. } \\ \text { Constant } & -0.9171 & 0.1249 & -7.34 & 0.000\end{array}\) \begin{tabular}{lrrrr} Constant & -0.9171 & 0.1249 & -7.34 & 0.000 \\ Distance to Crop Edge & -0.10716 & 0.01062 & -10.09 & 0.000 \\ \hline \end{tabular} \(\mathrm{S}=0.387646 \quad \mathrm{R}-\mathrm{Sq}=72.8 \% \quad \mathrm{R}-\mathrm{Sq}(\mathrm{adj})=72.1 \%\) a. What is the logistic regression function relating \(x\) and the proportion of plants with virus symptoms? b. What would you predict for the proportion of plants with virus symptoms at a distance of 15 meters from the edge of the field? (Note: the \(x\) values in the data set ranged from 0 to \(20 .)\)

The sales manager of a large company selected a random sample of \(n=10\) salespeople and determined for each one the values of \(x=\) years of sales experience and \(y=\) annual sales (in thousands of dollars). A scatterplot of the resulting \((x, y)\) pairs showed a linear pattern. a. Suppose that the sample correlation coefficient is \(r=.75\) and that the average annual sales is \(\bar{y}=100 .\) If a particular salesperson is 2 standard deviations above the mean in terms of experience, what would you predict for that person's annual sales? b. If a particular person whose sales experience is 1\. 5 standard deviations below the average experience is predicted to have an annual sales value that is 1 standard deviation below the average annual sales, what is the value of \(r\) ?
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