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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