Class 9: Thurs., Oct. 7

1
Class 9 Thurs., Oct. 7

• Inference in regression (Ch 10.1-10.2)
• Confidence intervals for slope
• Hypothesis test for slope
• Confidence intervals for mean response
• Prediction intervals
• Confidence intervals and the polls
• I will e-mail HW 5 to you by tomorrow. It will
  be due Tuesday, Oct. 19th.

2
CPS Wage-Education Data for March 1988
3
Inference Based on Sample

• The whole Current Population Survey (25,631 men
  ages 18-70) is a random sample from the U.S.
  population (roughly 75 million men ages 18-70).
• In most regression analyses, the data we have is
  a sample from some larger (hypothetical)
  population. We are interested in the true
  regression line for the larger population.
• Inference Questions
• How accurate is the least squares estimate of the
  slope for the true slope in the larger
  population?
• What is a plausible range of values for the true
  slope in the larger population based on the
  sample?
• Is it plausible that the slope equals a
  particular value (e.g., 0) based on the sample?
• Regression Applet
• Link on web site under Fun Links. Link
  entitled Simple Linear Regression.

4
Full Data Set
Random Sample of Size 25
5
Confidence Intervals

• Confidence interval A range of values that are
  plausible for a parameter given the data.
• 95 confidence interval An interval that 95 of
  the time will contain the true parameter.
• Approximate 95 confidence interval Estimate of
  parameter 2SE(Estimate of parameter).
• Approximate 95 confidence interval for slope
• For wage-education data,
  , approximate 95 CI
• Interpretation of 95 confidence interval It is
  most plausible that the true slope is in the 95
  confidence interval. It is possible that the
  true slope is outside the 95 confidence interval
  but unlikely the confidence interval will fail
  to contain the true slope only 5 of the time in
  repeated samples.

6
Conf. Intervals for Slope in JMP

• After Fit Line, right click in the parameter
  estimates table, go to Columns and click on Lower
  95 and Upper 95.
• The exact 95 confidence interval is close to but
  not equal to

7
Hypothesis Testing

• Simple Linear Regression Model
• Is the slope equal to 0?
• Null hypothesis
• Alternative (research) hypothesis
• Test statistic
• Rough rule Reject if tgt2. Accept
  if tlt2.
• P-values Find the p-value for the test. The
  p-value is a measure of the credibility of the
  null hypothesis. Small p-values give you
  evidence against the null hypothesis. Large
  p-values suggest there is no evidence in the data
  to reject the null hypothesis.
• The generally followed rule is to reject
  if the p-value is less than 0.05 and accept
  if the p-value is greater than 0.05.

8

• Hypothesis Testing in JMP
• The test statistic is a standard error counter.
  It is the relationship between and
  that matters, not the size of itself.
• Testing vs. .
  Use test statistic . Reject null
  hypothesis if tgt2.

9
(No Transcript)
10
Logic of Hypothesis Testing Hypoth. Testing in
the Courtroom

• Null hypothesis The defendant is innocent
• Alternative hypothesis The defendant is guilty
• The goal of the procedure is to determine whether
  there is enough evidence to conclude that the
  alternative hypothesis is true. The burden of
  proof is on the alternative hypothesis.
• A small p-value indicates that there is strong
  evidence against the null hypothesis. A p-value
  gt 0.05 does not show that the null hypothesis is
  true, only that there is not strong evidence
  against the null hypothesis.

11
Car Price Example

• A used-car dealer wants to understand how
  odometer reading affects the selling price of
  used cars.
• The dealer randomly selects 100 three-year old
  Ford Tauruses that were sold at auction during
  the past month. Each car was in top condition
  and equipped with automatic transmission, AM/FM
  cassette tape player and air conditioning.
• carprices.JMP contains the price and number of
  miles on the odometer of each car.

12
(No Transcript)
13

• The used-car dealer has an opportunity to bid on
  a lot of cars offered by a rental company. The
  rental company has 250 Ford Tauruses, all
  equipped with automatic transmission, air
  conditioning and AM/FM cassette tape players.
  All of the cars in this lot have about 40,000
  miles on the odometer. The dealer would like an
  estimate of the average selling price of all cars
  of this type with 40,000 miles on the odometer,
  i.e., E(YX40,000).
• The least squares estimate is

14
Confidence Interval for Mean Response

• Confidence interval for E(YX40,000) A range of
  plausible values for E(YX40,000) based on the
  sample.
• Approximate 95 Confidence interval
• Notes about formula for SE Standard error
  becomes smaller as sample size n increases,
  standard error is smaller the closer is to
• In JMP, after Fit Line, click red triangle next
  to Linear Fit and click Confid Curves Fit. Use
  the crosshair tool by clicking Tools, Crosshair
  to find the exact values of the confidence
  interval endpoints for a given X0.

15
(No Transcript)
16
A Prediction Problem

• The used-car dealer is offered a particular
  3-year old Ford Taurus equipped with automatic
  transmission, air conditioner and AM/FM cassette
  tape player and with 40,000 miles on the
  odometer. The dealer would like to predict the
  selling price of this particular car.
• Best prediction based on least squares estimate
  

17
Range of Selling Prices for Particular Car

• The dealer is interested in the range of selling
  prices that this particular car with 40,000 miles
  on it is likely to have.
• Under simple linear regression model, YX
  follows a normal distribution with mean
  and standard deviation . A car with
  40,000 miles on it will be in interval
  about 95 of the time.
• Class 5 We substituted the least squares
  estimates for
  for and said car with 40,000
  miles on it will be in interval
  about 95 of the time.
  This is a good approximation but it ignores
  potential error in least square estimates.

18
Prediction Interval

• 95 Prediction Interval An interval that has
  approximately a 95 chance of containing the
  value of Y for a particular unit with XX0 ,where
  the particular unit is not in the original
  sample.
• Approximate 95 prediction interval
• In JMP, after Fit Line, click red triangle next
  to Linear Fit and click Confid Curves Indiv. Use
  the crosshair tool by clicking Tools, Crosshair
  to find the exact values of the prediction
  interval endpoints for a given X0.

19
(No Transcript)
20
Comparison of Confidence Intervals for Mean
Response and Prediction Intervals

• Confidence Interval for Mean Response
• Prediction Interval
• Prediction interval is wider than confidence
  interval for mean response because it is trying
  to predict the Y for a particular unit with XX0
  rather than the mean for all units with XX0
• As sample size (n) becomes large, width of
  confidence interval for mean response goes to
  zero whereas width of prediction interval goes to
  2RMSE.

21
Confidence Intervals and the Polls

• Margin of Error 2SE(Estimate).
• 95 CI for Bush-Kerry difference
• 95 CI for difference between Bush and Kerrys
  proportions

22
Why Do the Polls Sometimes Disagree So Much?
23
Validity of Confidence Interval

• Polls are conducted by attempting to randomly
  sample U.S. citizens of voting age.
• Mean Estimated Difference in Vote Proportion
  Average Estimated Difference in Vote Proportion
  from repeated random samples.
• SE(Estimated Difference in Vote Proportion) is
  the typical amount by which the Estimated
  Difference in Vote Proprtion for one random
  sample differs from the Mean Estimated Difference
  in Vote Proportion
• CI for True Difference in Vote Proportion
  Estimated Difference in Vote Proportion
  2SE(Estimated Difference in Vote Proportion)
• Confidences interval 95 guarantee that 95 of
  the time it will contain true difference in vote
  proportion is only true if mean estimated
  difference in vote proportion true difference
  in vote proportion.
• When mean estimated difference in vote proportion
  does not equal true difference in vote
  proportion, there is bias.

24
Sources of Bias

• See Ch 3.3 pages 252-254
• Undercoverage some groups in the population are
  left out of the process of choosing the sample
  (for an opinion poll conducted by telephone,
  people without a residential phone are not
  covered).
• Nonresponse An individual chosen for the sample
  cant be contacted or does not cooperate. Major
  problem in telephone surveys.
• Response bias Respondents or interviewers
  behavior may cause bias. Respondents may lie,
  especially if asked about illegal or unpopular
  behavior. Race or sex of interviewer can affect
  responses.
• Wording of questions Has very important
  influence on survey results. UN Experiment.

25
Voting Polls

• The polls try to predict what will happen in the
  election. Thus, they must address the question,
  who is likely to vote?
• In exit polling from the 2000 election, 39 of
  respondents identified themselves as Democrats,
  34 as Republicans and 33 as Independents. In
  1996, the composition was 39 Democrat, 34
  Republican and 27 Independent.
• Should the polls adjust their results so that
  they reflect a voter composition of more
  Democrats than Republicans?
• Gallup doesnt do much adjustment.
• LA Times poll, Zogbys poll make adjustments.