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Chapter 5: Problem 76

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

Short Answer

Expert verified
The sample correlation coefficient 'r' is 0. When the fifth observation is made at x=6, y=8 is a value for which 'r' > 0.5 and y=-8 is a value for which r < 0.5.

Step by step solution

01

Calculation of the Sample Correlation Coefficient 'r'

First, we use the formula for 'r': \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \] For the four given points, the sums are calculated as follows: \(n = 4, \sum x = 4, \sum y = 0, \sum xy = 0, \sum x^2 = 6, \sum y^2 = 2\). Substituting these values into the formula gives us: \[ r = \frac{4(0) - (4)(0)}{\sqrt{[4*6 - (4)^2][4*2 - (0)^2]}} = 0 \]
02

Finding a value of 'y' to increase 'r' beyond 0.5

We need r > 0.5, thus we want a value of y when x=6 that increases the correlation. Because 'r' increases when 'x' and 'y' increase together, we should choose a positive value for 'y'. There are different ways to find an exact number, one could use trial and error or use software. By trying different integers, we find y=8 works. So, when x=6, y=8, r > 0.5.
03

Finding a value of 'y' to decrease 'r' below 0.5

We need r < 0.5, thus we want a value of y when x=6 that decreases the correlation. Because 'r' decreases when 'x' increases and 'y' decreases, we should choose a negative value for 'y'. There are different ways to find an exact number for 'y', one could use trial and error or use software. By trying different integers, we find y=-8 works. So, when x=6, y=-8, r < 0.5.

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Key Concepts

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

Statistical Analysis
Statistical analysis entails collecting, reviewing, interpreting, and presenting data to discover patterns and trends. A fundamental tool in this field is the sample correlation coefficient, often denoted as 'r'. This coefficient measures the strength and direction of a linear relationship between two variables on a scatter plot. Values of 'r' range from -1 to +1. An 'r' value of +1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no linear correlation. Understanding 'r' helps in predicting trends and making decisions based on data patterns. It's important to note that while correlation implies association, it does not signify causation. Statistical analysis requires careful consideration of the context and limitations of the data when interpreting these coefficient values.
Correlation Calculation
The calculation of the sample correlation coefficient 'r' is an essential process within statistics. It involves applying a specific formula where data points, such as \( (x, y) \) pairs, are plugged into an equation. In the exercise scenario, with the given pairs, the formula for 'r' takes into account the sum of the products of \( x \) and \( y \) for all pairs, the sum of \( x \) values, the sum of \( y \) values, as well as the sums of the squares of \( x \) and \( y \) values.

The result, which in the provided example is 0, indicates no linear relationship among the initial four pairs. This understanding of how each component affects the calculation is crucial for students who might need to manipulate the dataset, for instance by adding a new \( (x, y) \) pair, to achieve a desired correlation as seen in subsequent steps of the exercise.
Data Interpretation
Data interpretation involves making sense of the numerical findings from statistical analysis. It requires critical thinking to discern what the numbers represent in a given context. For the sample correlation coefficient, the interpretation aids in understanding the relationship between the datasets. In a scenario with an 'r' value of 0, it means no linear relationship exists between the variables being studied.

However, when new data is introduced, as with the fifth observation in the exercise, interpretation involves predicting the effect of this new data point on the existing correlation. For instance, adding an observation where \( x=6 \) and searching for a \( y \) value that correlates to 'r' being more or less than 0.5 requires the ability to anticipate the impact of \( y \) values on the coefficient. Choosing positive values of \( y \) increases the correlation, while negative values decrease it, exemplifying how the interpretation of data aligns with the mathematical calculations to drive conclusions.

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Most popular questions from this chapter

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