Ch 3 Two dimension concept
Ch 4 Two dimensions concept
I. Scatter plot
Linear Quadratic Power Exponential
regression regression regression regression
e
Perfect Positive Negative No
correlation correlation correlation correlation
As x increases, As x increases,
y increases y decreases
II. Correlation coefficient (r)
1. Covariance between X and Y
The covariance is a measure of how two variables vary together.
2. When = 0, there is no correlation.
If is large, it is difficult to interpret.
Pearson used a formula to standardize the covariance, which is called the Pearson Correlation Coefficient.
which is equivalent to
where [pic] is the sample standard deviation of the explanatory variable [pic],
[pic] is the sample standard deviation of the response variable [pic].
3. [pic] measures the strength of the linear association between X and Y.
4. [pic]
r = -1 r = 0 r = 1
Perfect negative No correlation Perfect positive
correlation correlation
If [pic] > the critical value of correlation coefficient obtained from Table II,
there is a positive/ negative linear correlation between [pic]and [pic] .
III. Regression Line
If there is positive / negative correlation between X and Y, find the best fitted line for the data.
The least-squares regression line, [pic], is the line that minimizes the sum of the squared errors (residuals).
Residual = observed [pic]- predicted [pic]
= [pic]
The least squares regression line is [pic] where [pic] is the slope of the least squares regression line and [pic] is the [pic]-intercept
of the least squares regression line.
Summary:
1. Use StatCrunch to plot a scatter plot
2. Use StatCrunch to calculate [pic].
3. Determine whether there is a positive/negative linear correlation
between X and Y.
If > the critical value of Correlation Coefficient obtained from
Table II, there is a positive linear correlation between X and Y for r
is positive and there is a negative linear correlation between X and Y
for r is negative.
4. If there is a linear correlation between X and Y, use StatCrunch to find
the least squares regression line. Otherwise, do not find the least squares regression line.
5. When a value is assigned to X --> if there is a correlation between X and Y, use the least squares regression line to find the best predicted Y.
When a value is assigned to X--> if there is no correlation between X and Y, use StatCrunch to find [pic]and the best predicted Y is [pic] for any X.
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