LINEAR REGRESSION AND CORRELATION EXTENDED
LINEAR REGRESSION AND CORRELATION EXTENDED
Recall that sample linear correlation was
[pic]
There is a population linear correlation too, it is denoted by [pic](rho). This is a population number and we will not give a formula for it since it is unlikely you would have data for an entire population. However the reason for calculating r is to have a guess about [pic].
HTs for
Case 1: Ho: [pic] (no correlation) Ha: [pic] (some correlation) (this is a 2-tail test)
Case 2: Ho: [pic] (not a + correlation) Ha: [pic] (+ correlation) (this is a right tail test)
Case 3: Ho: [pic](not a – correlation) Ha: [pic] (- correlation) (this is a left tail test)
Use [pic]=[pic]with df = n-2 .
If there is evidence that [pic] , i.e. there is some correlation, positive or negative and the following are true then it makes sense to find the line of best fit that we talked about before.
• data is linear
• data is independent
• data is normally distributed about the line
• standard deviation of the data about the line is same all along the line
The last two are hard to check unless you have a very large data set. You should also make a scatter plot graph because outliers have large affect.
Before we made our best guess for a y based on a particular value of x using the line of best fit. Now we turn this guess into a CI. There are two cases.
CI for average of all y’s with [pic]:
[pic]
CI for a single value of y given a value of [pic]:
[pic]
[pic]is obtained by the regression line [pic] with [pic]in place of x. [pic]is the best guess that we did before based on the x that we are interested in which is the [pic].
[pic]is the critical value off the t-table with n-2 degrees of freedom.
We will give some insight into why the df=n-2. We are seeing how points are scattered about a line and it takes two points to fix a line, so in a sense after you do this the other n-2 points can then be compared to the line.
It should be noted that [pic]can be very sensitive to rounding.
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