2007 (C) Nancy Pfenning
Looking Back: Review
Lecture 28 Categorical & Quantitative Variable Inference in Paired Design
Inference
4 Stages of Statistics
Data Production (discussed in Lectures 1-4) Displaying and Summarizing (Lectures 5-12) Probability (discussed in Lectures 13-20) Statistical Inference
for Relationships: 2 Approaches CatQuan Relationship: 3 Designs Inference for Paired Design Paired vs. Ordinary, t vs. z
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture
1 categorical (discussed in Lectures 21-23) 1 quantitative (discussed in Lectures 24-27) cat and quan: paired, 2-sample, several-sample 2 categorical 2 quantitative
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.2
Inference for Relationships: Two Approaches
Example: CQ Test Relationship or Parameters
and and
about variables: not related or related about parameters: equality or not
Applies to all three CQ, CC, QQ CQ: pop means equal? (mean diff=0? for paired) CC: pop proportions equal? QQ: pop slope equals zero?
Background: Research question: For all students at a university, are their Math SATs related to what year theyre in? Question: How can we formulate this in terms of parameters?
Either way, often do test before confidence interval. 1. Are variables related? 2. If so, quantify: how different are the parameters?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L28.3 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L28.4
Elementary Statistics: Looking at the Big Picture
1
(C) 2007 Nancy Pfenning
Example: CQ Test Relationship or Parameters
Design for CatQuan Relationship (Review)
Background: Research question: For all students at a university, are their Math SATs related to what year theyre in? Response:
Paired Two-Sample Several-Sample
Looking Ahead: Inference procedures for population relationship will differ, depending on which of the three designs was used.
Looking Ahead: This is a several-sample design, to be discussed after paired and two-sample.
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.6
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.7
Inference Methods for CatQuan Relationship
Example: Paired vs. Two-Sample Data
Paired: reduces to 1-sample t (already covered) Two-Sample: 2-sample t (similar to 1-sample t) Several-Sample: need new distribution (F)
Background: Research Question: Are age of parent and sex of parent related for population of students at a university? Question: How can this data set be used to answer the research question?
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.8
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.9
Elementary Statistics: Looking at the Big Picture
2
(C) 2007 Nancy Pfenning
Example: Paired vs. Two-Sample Data
Paired Data: Incorrect vs. Correct Approach
Background: Research Question: Are age of parent and sex of parent related for population of students at a university? Response:
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.11
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.12
Example: Paired vs. Two-Sample Summary
Example: Paired vs. Two-Sample Summary
Background: Research Question: Are age of parent and sex of parent related for population of students at a university? Question: Which output has enough information to perform inference?
Background: Research Question: Are age of parent and sex of parent related for population of students at a university? Response:
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.13
Looking Ahead: We will standardize with the StDev of the differences, which cannot be found from the individual StDevs because of dependence. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture
L28.15
Elementary Statistics: Looking at the Big Picture
3
(C) 2007 Nancy Pfenning
Example: Consider Summaries in Paired Design
Example: Consider Summaries in Paired Design
Background: To see if age of parent and sex of parent are related for population of students at a university, took sampled DadAge minus MomAge. Question: Which parent tended to be older in the sample?
Background: To see if age of parent and sex of parent are related for population of students at a university, took sampled DadAge minus MomAge. Response: Mean of diffs >0_____ older
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.16
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.18
Example: Display in Paired Design
Example: Display in Paired Design
Background: To see if age of parent and sex of parent are related for population of students at a university, took sampled DadAge minus MomAge. Question: How do we display the data?
Background: To see if age of parent and sex of parent are related for population of students at a university, took sampled DadAge minus MomAge. Response: Construct histogram of differences.
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.19
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.20
Elementary Statistics: Looking at the Big Picture
4
(C) 2007 Nancy Pfenning
Example: Display in Paired Design
Example: Display in Paired Design
Background: Histogram of age differences:
Background: Histogram of age differences:
Question: What does the histogram show?
Response: Age differences have
Center: around ____(dads tend to be about___ yrs older) Spread: most diffs within _______ yrs of mean Shape: ___________ (a few dads much older than wife)
Elementary Statistics: Looking at the Big Picture L28.23
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.21
(C) 2007 Nancy Pfenning
Notation in Paired Study
Test Statistic in Paired Study
Differences have
Sample mean Population mean Sample standard deviation Population standard deviation
Start with ordinary 1-sample statistic Substitute Substitute 0 for ( for ordinary summaries will claim )
Result is paired t statistic:
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.24
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.25
Elementary Statistics: Looking at the Big Picture
5
(C) 2007 Nancy Pfenning
Example: Paired t Test
Example: Paired t Test
Background: Paired test on students parents ages:
Background: Paired test on students parents ages:
Question: What does output tell about formal test?
Response: Testing
: ______ vs.
: _______
Sample unbiased? ___ n=431 large? ___ Pop4310? ___ Large? _____ P-value = _____. Small? ____ Conclude pop mean diff =0?___Sex and age related?___
Elementary Statistics: Looking at the Big Picture L28.28
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.26
(C) 2007 Nancy Pfenning
Example: One- or Two-Sided
in Paired Test
Example: or One- Two-Sided
in Paired Test
Background: Paired test on students parents ages:
Background: Paired test on students parents ages:
Question: What would change if wed suspected students fathers tend to be older than mothers?
Response: Replace
with _________
P-value would be __________________ Conclude fathers in general are older? ______
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.29
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.31
Elementary Statistics: Looking at the Big Picture
6
(C) 2007 Nancy Pfenning
Example: Paired vs. Ordinary t vs. z
Example: Paired vs. Ordinary t vs. z
Background: Paired test on 431 students parents ages resulted in paired t-statistic +13.11. Question: What does this tell us about the P-value?
Background: Paired test on 431 students parents ages resulted in paired t-statistic +13.11. Response:
Paired t same as ordinary t distribution Ordinary t basically same as z for large n 13.11 sds above mean unusual? ____ P-val _____ Evidence that mean age diff is non-zero in pop.?____ Note: for extreme t statistics, software not needed to estimate P-value.
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.32
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.34
C.I. for Mean:
Unknown (Review)
is
Confidence Interval in Paired Design
Confidence interval for is
95% confidence interval for
multiplier from t distribution with n-1 degrees of freedom (df) multiplier at least 2, closer to 3 for very small n
Multiplier from t distribution with n-1 df Multiplier smaller for lower confidence Multiplier smaller for larger df If n is small, diffs need to be approx. normal. (Same guidelines as for 1-sample t)
L28.35 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L28.36
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture
7
(C) 2007 Nancy Pfenning
Guidelines: Sample Mean Diff Approx. Normal Can assume shape of for random samples of n pairs is approximately normal if Graph of sample diffs appears normal; or Graph of sample diffs fairly symmetric and n at least 15; or Graph of sample diffs moderately skewed and n at least 30; or Graph of sample diffs very skewed and n much larger than 30
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L28.37
Example: Paired Confidence Interval
Background: Sample of 431 students parents age differences have mean +2.45, s.d. 3.88. Question: What is a 95% confidence interval for population mean age difference?
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.38
Example: Paired Confidence Interval
Example: Checking Conditions for Paired t
Background: Sample of 431 students parents age differences have mean +2.45, s.d. 3.88. Response: Since n is so large, t multiplier same as z: 2 for 95% confidence. (Also, skewed hist. OK.) Pretty sure that fathers are older by about 2.1 to 2.8 years in population from which sample was taken.
Background: Mileages for 5 cars, each tested in city and on highway (suspect higher on highway).
Question: Is paired t procedure appropriate?
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.40
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.41
Elementary Statistics: Looking at the Big Picture
8
(C) 2007 Nancy Pfenning
Example: Checking Conditions for Paired t
Example: Paired Test and Confidence Interval
Background: Mileages for 5 cars, each tested in city and on highway (suspect higher on highway).
Background: Mileages for 5 cars, each tested in city and on highway (suspect higher on highway).
Response: Histogram _________marginally OK
Question: What does the output tell us?
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.43
(C) 2007 Nancy Pfenning
Elementary Statistics: Looking at the Big Picture
L28.44
Example: Paired Test and Confidence Interval
Example: Paired Confidence Interval by Hand
Background: Mileages for 5 cars, each tested in city and on highway (suspect higher on highway).
Background: Mileage differences for ...