COMMERCE 2QA3 Study Guide - Final Guide: Test Statistic, Confidence Interval, Probability Plot
28 Jun 2018
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Department
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Perform a Hypothesis Test, Calculate the CI for the Slope of the Linear Regression
The Population and the Sample
Usually we model relationships by fitting a straight line to a sample of ordered pairs (the line of best fit).
Observations differ from sample to sample and as a result,
.
The True Line that summarizes the relationship (of x and y) for the entire population
This occurs in an idealized case where the points are perfectly linear
•
may not lie on the line , but be distributed around it - this error is accounted for by adding , such
that
•
where is the population mean.
Regression Inference
The residuals
are the sample based versions of
•
You can account for uncertainties in and by making confidence intervals
•
Estimate the by finding the regression line, where
. Note that .
Assumptions and Conditions (Check in this order)
Linearity
Assumption
Make a scatterplot of the data.
Independence
Assumption
Fit a regression and find the residuals and predicted values, then plot the residuals
against time and check for evidence of patterns.
Equal Variance
Assumption
Make a scatterplot of the residuals against or the predicted values - this plot should not
have a fan or cone shape.
Normal Population
Assumption
Make a histogram and Normal probability plot of the residuals. If distributions line up
along the line of best fit and have the same standard deviations, they follow the Normal
model.
Standard Error of the Slope
The standard error of the slope is a measure of variability of about the true slope .
Spread around the line which increases
•
Spread of x-values which decreases
•
Sample size which decreases
•
The standard error of the regression slope, is affected by ONLY three things:
The standardized estimated regression slope
follows the student's t-model with degrees
of freedom, the equation above is used
•
is the number of data values and is the standard deviation of the x-values
•
Standard error of the slope
Spread around the line
t-Test for Regression Slope
When assumptions and conditions are met, we can test the hypothesis vs. using the
standardized estimated regression slope. The t-model is used to find the p-value of the test, most often used
with to see if the slope is significantly different from zero.
Confidence Interval for the Regression Slope
Confidence Interval for the Regression
Slope
where the critical value depends on the
confidence level, and has degrees of freedom
Inference for Regression
November 2, 2017
4:34 PM
Statistics Page 1
Document Summary
Perform a hypothesis test, calculate the ci for the slope of the linear regression. The true line that summarizes the relationship (of x and y) for the entire population. Usually we model relationships by fitting a straight line to a sample of ordered pairs (the line of best fit). Observations differ from sample to sample and as a result, : where is the population mean. This occurs in an idealized case where the points are perfectly linear: may not lie on the line , but be distributed around it - this error is accounted for by adding , such that. Estimate the by finding the regression line, where . The residuals are the sample based versions of. You can account for uncertainties in and by making confidence intervals. Fit a regression and find the residuals and predicted values, then plot the residuals against time and check for evidence of patterns.
