Statistical Sciences 2244A/B Lecture Notes - Lecture 15: Null Hypothesis, Confidence Interval, Statistical Hypothesis Testing
22 May 2018
School
Department
Professor
Stats 2244
Regression
Hypothesis test for slope parameter
Hypothesis test for slope (i.e β )
- The only regression model that we use in this course is a linear model
o We are trying to describe the relationship bw two quantitative variables which is linera
- Bivariant data: collecting data about 2 different variables (and so the relationship bw the
variables)
- There are diff hypothesis tests you can do to find the relationship bw the two variables
o The hypothesis test for the slope parameter (symbol used for this is beta)
- null hypotheses has the same structure
o B = 0 would mean there is no relationship
o B0 is just a placeholder for whatever that null value is
- We can use this to test if there is a linear relationship →does y change with x in a linear way?
- A relationship with a slope of 0 doesn’t even have a relationship
- If we reject the hypothesis that B=0, then that means that there is some relationship
- Beta is the real slope if we were in the population and had all the data in the population
- This equation is an estimate on how x and y are related
- NOTE: THE DEGREES OF FREEDOM IS N-2
Clicker Question
Conditions for test of slope parameter/linearity
- There are 4 conditions for this hypothesis test and the corresponding confidence interval
1. Y observations are independent
o Are the individual values of y independent?
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o If we are looking at the relationship of leaf size (explanatory variable) and the number
of leves (response variable) →is there is a relationship bw these 2 variables?
2. Linear relationship bw Y and X
o If there was a relationship, a line would be a good descriptor of it, then we would be able
to go and test what the value of that line is
3. Y follows a normal distribution for each x
o Unimodal, symmetric, bell shaped
o QQ plot
o For each x means
4. Constant variance of Y (and ) across X
o Recall: variance is a measure of spread
- these conditions arise from the model describing the relationship between Y and X
- these conditions must be checked when we want to do the hypothesis test and they are based on
the underlying model of regression
Model: Describing Y based on x
o goal: describe steves weight based on his height
o we are working with a linear equation, and so there is a slope and a y intercept
o this is what alpha and beta are
o but alpha and beta are describing the relationship in the population not the sample data
o alpha = y intercept
o beta = slope
o note: this linear line is running through the mean of those 3 normal distributions
o this means that alpha +betax is a meave value of y for a given value of x
o steve is represented by the * below
o he is below the mean for the normal distribution at 185cm
o this is where error term comes in →difference bw steves weight and the mean of people
who have his height
Residual (y – y hat) plots
- something that is nonlinear will have a weird looking residual plot
find more resources at oneclass.com
find more resources at oneclass.com
