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Chapter 4: Problem 93

Construct a set of numbers (with at least three points) with a strong negative correlation. Then add one point (an influential point) that changes the correlation to positive. Report the data and give the correlation of each set.

Short Answer

Expert verified
An initial data set might consist of points (1,10), (2,8), and (3,6) creating a negative correlation. Adding an influential point such as (10,20) to this set will change the correlation to positive.

Step by step solution

01

Creation of a set of numbers with strong negative correlation

We first create a set of numbers (x, y) such as (1,10), (2,8), and (3,6). If we plot these points, we can visibly see a negative correlation; as x increases, y decreases.
02

Calculation of correlation for the initial set

Using a statistical calculation tool or calculator, we compute the correlation coefficient \( r \) for the set of points. It should be close to -1, implying a strong negative correlation.
03

Addition of an influential point

We add a point to the data that will shift the correlation from negative to positive. To perform this, we need to insert a point that has a much higher value in both x and y. For instance, let's add the point (10, 20). This point is higher than any other points in both x and y.
04

Calculation of correlation for the new set

Again, we calculate the correlation coefficient \( r \) for the new set of points. Now, the coefficient should be positive and closer to +1, reflecting a strong positive correlation.

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Key Concepts

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

Negative Correlation
Negative correlation occurs when two variables move in opposite directions. As one variable increases, the other decreases. This relationship can be visualized as a downward-sloping line on a graph.
For example, consider the data points (1,10), (2,8), and (3,6). When plotted, they show that as the x-values (1, 2, 3) increase, the y-values (10, 8, 6) decrease.
To quantify this negative relationship, we calculate the correlation coefficient. For a strong negative correlation, this value will be close to -1, signifying that as one variable rises, the other falls almost perfectly.
Positive Correlation
Positive correlation describes a situation where two variables move in the same direction. When one variable increases, the other also increases. This results in an upward-sloping line on a graph.
Imagine you start with a dataset showing negative correlation and then add a new point, like (10, 20), with significantly higher x and y values. This point can alter the overall trend to show a positive correlation.
By recalculating the correlation coefficient after adding the influential point, you may get a positive value closer to +1, indicating a strong positive relationship.
Influential Point
An influential point is an outlier that significantly changes the correlation between two variables. It's a data point that stands out due to its distinct values, affecting the slope of the trendline.
In the context of our exercise, adding (10, 20) to the initial points (1,10), (2,8), and (3,6) acts as an influential point. It drastically alters the correlation from negative to positive, showing how a single point can sway analytical results.
Identifying influential points is crucial for data analysis as they can mislead interpretations or reveal new patterns.
Correlation Coefficient
The correlation coefficient, denoted as r, is a statistical measure that describes the direction and strength of a linear relationship between two variables.
The value of r ranges from -1 to +1.
  • r = +1 means a perfect positive correlation.
  • r = -1 means a perfect negative correlation.
  • r = 0 implies no correlation.
For example, at first, the data points (1,10), (2,8), and (3,6) have a negative correlation coefficient close to -1. After adding the influential point (10,20), recalculating gives a correlation coefficient that changes, demonstrating its sensitivity to data alterations.

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Most popular questions from this chapter

Data on the number of home runs, strikeouts, and batting averages for a sample of 50 Major League Baseball players were obtained. Regression analyses were conducted on the relationships between home runs and strikeouts and between home runs and batting averages. The StatCrunch results are shown below. (Source: mlb.com) Simple linear regression results: Dependent Variable: Home Runs Independent Variable: Strikeouts Home Runs \(=0.092770565+0.22866236\) Strikeouts Sample size: 50 \(\mathrm{R}\) (correlation coefficient) \(=0.63591835\) \(\mathrm{R}-\mathrm{sq}=0.40439215\) Estimate of error standard deviation: \(8.7661607\) Simple linear regression results: Dependent Variable: Home Runs Independent Variable: Batting Average Home Runs \(=45.463921-71.232795\) Batting Average Sample size: 50 R (correlation coefficient) \(=-0.093683651\) \(\mathrm{R}-\mathrm{sq}=0.0087766264\) Estimate of error standard deviation: \(11.30876\) Based on this sample, is there a stronger association between home runs and strikeouts or home runs and batting average? Provide a reason for your choice based on the StatCrunch results provided.

In Exercise \(4.1\) there is a graph of the relationship between SAT score and college GPA. SAT score was the predictor and college GPA was the response variable. If you reverse the variables so that college GPA was the predictor and SAT score was the response variable, what effect would this have on the numerical value of the correlation coefficient?

Coefficient of Determination If the correlation between height and weight of a large group of people is \(0.67\), find the coefficient of determination (as a percentage) and explain what it means. Assume that height is the predictor and weight is the response, and assume that the association between height and weight is linear.

Construct a set of numbers (with at least three points) with a strong positive correlation. Then add one point (an influential point) that changes the correlation to negative. Report the data and give the correlation of each set.

The following table shows the number of text messages sent and received by some people in one day. (Source: StatCrunch: Responses to survey How often do you text? Owner: Webster West. A subset was used.) a. Make a scatterplot of the data, and state the sign of the slope from the scatterplot. Use the number sent as the independent variable. b. Use linear regression to find the equation of the best-fit line. Graph the line with technology or by hand. c. Interpret the slope. d. Interpret the intercept. $$ \begin{array}{|c|c|} \hline \text { Sent } & \text { Received } \\ \hline 1 & 2 \\ \hline 1 & 1 \\ \hline 0 & 0 \\ \hline 5 & 5 \\ \hline 5 & 1 \\ \hline 50 & 75 \\ \hline 6 & 8 \\ \hline 5 & 7 \\ \hline 300 & 300 \\ \hline 30 & 40 \\ \hline \end{array} $$ $$ \begin{array}{|r|r|} \hline \text { Sent } & \text { Received } \\ \hline 10 & 10 \\ \hline 3 & 5 \\ \hline 2 & 2 \\ \hline 5 & 5 \\ \hline 0 & 0 \\ \hline 2 & 2 \\ \hline 200 & 200 \\ \hline 1 & 1 \\ \hline 100 & 100 \\ \hline 50 & 50 \\ \hline \end{array} $$
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