Inference for Regression

1
Lesson 15 - 1

• Inference for Regression

2
Knowledge Objectives

• Identify the conditions necessary to do inference
  for regression.
• Explain what is meant by the standard error about
  the least-squares line.

3
Construction Objectives

• Given a set of data, check that the conditions
  for doing inference for regression are present.
• Compute a confidence interval for the slope of
  the regression line.
• Conduct a test of the hypothesis that the slope
  of the regression line is 0 (or that the
  correlation is 0) in the population.

4
Vocabulary

• Statistical Inference tests to see if the
  relationship is statistically significant

5
Conditions for Regression Inference

• Repeated responses y are independent of each
  other
• The mean response, µy, has a straight-line
  relationship with x
  µy a ßxwhere the slope ß and
  intercept a are unknown parameters
• The standard deviation of y (call it s) is the
  same for all values of x. The value of s is
  unknown.
• For any fixed value of x, the response variable y
  varies according to a Normal distribution

6
Sampling Distribution Concepts

• Remember from our sampling distribution lesson
  how repeated samplings of the mean will be
  Normally distributed (n gt 30, CLT applies)

7
Checking Regression Conditions

• Observations are independent
• No repeated observations on the same individual
• The true relationship is linear
• Scatter plot the data to check this
• Remember the transformations to make non-linear
  data linear
• Response standard deviation is the same
  everywhere
• Check the scatter plot to see if this is violated
• Response varies Normally about the true
  regression line
• To check this, we look at the residuals (since
  they must be Normally distributed as well) either
  with a box plot or normality plot
• These procedures are robust, so slight departures
  from Normality will not affect the inference

8
Estimating the Parameters

• We need to estimate parameters for µy a ßx
  and s
• From the least square regression line y-hat a
  bx we get unbiased estimators a (for a) and b
  (for ß)
• We use n 2 because we used a and b as estimators

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Confidence Interval on ß

• Remember our form Point Estimate Margin of
  Error
• Since ß is the true slope, then b is the point
  estimate
• The Margin of Error takes the form of t ? SEb

10
Confidence Intervals in Practice

• We use rarely have to calculate this by hand
• Output from Minitab

Parameters b (1.4929), a (91.3), s (17.50)
t 2.042 from n 2, 95 CL
CI PE MOE 1.4929 (2.042)(0.4870)
1.4929 0.9944
0.4985, 2.4873
Since 0 is not in the interval, then we might
conclude that ß ? 0
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Inference Tests on ß

• Since the null hypothesis can not be proved, our
  hypotheses for tests on the regression slope will
  beH0 ß 0 (no correlation between
  x and y)Ha ß ? 0 (some linear
  correlation)
• Testing correlation makes sense only if the
  observations are a random sample.
• This is often not the case in regression
  settings, where researchers often fix in advance
  the values of x being tested

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Test Statistic
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Beer vs BAC Example

• 16 student volunteers at Ohio State drank a
  randomly assigned number of cans of beer. Thirty
  minutes later, a police officer measured their
  BAC. Here are the data
• Enter the data into your calculator.
• Draw a scatter plot of the data and the
  regression line
• Conduct an inference test on the effect of beers
  on BAC

Student 1 2 3 4 5 6 7 8
Beers 5 2 9 8 3 7 3 5
BAC 0.10 0.03 0.19 0.12 0.04 0.095 0.07 0.06
Student 9 10 11 12 13 14 15 16
Beers 3 5 4 6 5 7 1 4
BAC 0.02 0.05 0.07 0.10 0.085 0.09 0.01 0.05
LinReg(a bx) L1, L2, Y1
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Scatter plot and Regression Line
D F S O C

• Interpret the scatter plot

15
Output from Minitab

• Could we have used this instead of output from
  our calculator?

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Using the TI for Inference Test on ß

• Enter explanatory data into L1
• Enter response data into L2
• Stat ? Tests ? ELinRegTTest
• Xlist L1
• Ylist L2
• (Test type) ß ? ? 0 lt0 gt0
• RegEq (leave blank)
• Test will take two screens to output the
  dataInference t-statistic, degrees of freedom
  and p-valueRegression a, b, s, r², and r

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TI Output from page 907

• y a bx
• ß ? 0 and ? ? 0
• t 3.06548
• p .004105
• df 36
• a 91.26829
• b 1.492896
• s 17.49872
• r2 .206999
• r .4549725

Minitab Output
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Interpreting Computer Output

• In the following examples of computer output from
  commonly used statistical packages
• Find the a and b values for the regression eqn
• Find r and r2
• Find SEb, t-value and p-value (if available)
• We can use these outputs to finish an inference
  test on the association of our explanatory and
  response variables.

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Sample from Excel prob 15.10
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Sample from CrunchIt prob 15.20
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Summary and Homework

• Summary
• Inference Conditions Needed1) Observations
  independent2) True relationship is linear3) s
  is constant4) Responses Normally distributed
  about the line
• Confidence Intervals on ß can be done
• Inference testing on ß use the t statistic
  b/SEb
• Homework
• Pg 914 918 15.18-19, 15.21-23