Linear Relationships in Data
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1. Assume that a sample of n pairs of data results in the given value of r. For alpha equal .05, determine if there is a significant linear relationship between x and y. Let n= 22, r = 0.087 (yes or no)
2. Assume that a sample of n pairs of data results in the given value of r. For alpha equal .05 determine if there is a significant linear relationship between x and y. Let n = 40. r = -0.299 (yes or No)
3. Given the following ordered data pairs (x, y), compute the value of the linear correlation coefficient r: (2,6) (3,0) (5,15) (5,5) (10, 2). Give the value to 3 significant digits.
4. Given the following ordered pairs(x, y): (5,-2), (3,0) (2,1) (1,2)(0,3)(2,1), compute the equation of the regression line. Just type in the right side of the equation, and use 1 decimal place of accuracy for any numeric values.
5. A Pearson's Correlation coefficient between college entrance exam grades and scholastic achievement was computed on a very sophisticated computer program to be -1.08. The USAF has asked you to comment on the researchers methodology and findings. On the basis of your brief study, you would tell the USAF that:
Please show me the long-handed version using the formulas
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This solution provides equations and explanations for various questions involving linear relationships of data.
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QUESTIONS 9.
1. Assume that a sample of n pairs of data results in the given value of r. For alpha equal .05, determine if there is a significant linear relationship between x and y. Let n= 22, r = 0.087 (yes or no)
When truing to determine a linear relationship, we need to look at r - or the Pearson correlation. This data shows that the r = 0.087.
Remember that r can vary from -1 to +1, with a 0 in the middle. 0 means that there is NO correlation at all between the numbers. -1 means that there is perfect negative correlation (i.e 2 relationships are inversed - the more you workout, the more weight you loose, the more you sleep, the less tired you are..), and a +1 relationship means there is positive correlation (the more you study, the higher your grades).
The more the r deviates from 0, the stronger the relationship. A r=0.9 is a very strong positive relationship, while r=-0.9 is a very strong negative relationship. But we need to quantify this.
Here are some things to keep in mind:
1) The strength of the relationship between 2 variables:
? is indicated by the correlation coefficient: r
? but is actually measured by the coefficient of determination: r2
Testing for the significance of the correlation coefficient, r
So the first step to answer this question it so set up a null hypothesis:
Null hypothesis: r xy = 0.0
The simplest formula for computing the appropriate t value to test significance of a correlation coefficient employs the t distribution:
T = r (the square root of (n-2/1-r2))
o The degrees of freedom for entering the t-distribution is N - 2 = 20
T = 0.087 (the square root of (20/1-0.00757))
= 0.087(sq root of 20.152)
= 0.087(4.489)
T = ...
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