1. Stat 217 – Day 28
Review
Stat 217

2. Last Time – Paired data
When have paired observations (rather than independent samples) must analyze as paired data
Statistic: mean difference in sample
Parameter: mean difference in population
H0: m = 0
Theory-based approach: one-sample t-test
At least 20 differences or differences follow a normal distribution
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3. Last Time – Paired Data
Why might I choose to conduct a paired-data study?
Hope there is less variation in the differences than in the response variable itself (ages at marriage vs. age differences at marriage)
When can’t I conduct a paired-data study
Don’t have access to 2nd observation!
May not be natural pairings (e.g., students vs. faculty)
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4. Since first exam: Comparing 2 Groups
Categorical Response
Quantitative Response
Difference in sample proportions
Difference in population proportions/probabilities
Test of significance
Simulation (Two Proportions applet)
Normal distribution (TBI)
Confidence interval
Theory-Based applet
Difference in sample means
Difference in population means m1- m2
Test of significance
Simulation (Multiple Means applet)
t distribution (TBI)
Confidence interval
Theory-Based applet
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5. Including Paired Data Matched Pairs - Categorical
Matched Pairs - Quantitative
Sample proportion (without ties)
Population proportion p (of those that differ)
Test of significance
Simulation (One Prop)
Normal distribution (Theory-Based applet)
Confidence interval
Theory-Based applet
Sample mean difference
Population mean difference (mdiff)
Test of significance
Simulation (Matched Pairs Randomization – Lab 6)
Paired t test (one sample t test on differences)
Confidence interval
Theory-Based applet
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6. Format of Exam
Can bring two (two-sided) pages of notes that either I gave you or you self-produced
Prepare as if closed book
Work efficiently
Still bring calculator
Be able to read computer output
Be able to carry out/interpret simulations in Two Proportions, Multiple Means applets
Be able to describe simulation process
Be able to carry out/interpret theory-based procedures in Theory-Based Inference applet
Blue notes on review handout
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7. Choice of procedure
Is the question asking for a test of significance (“is there evidence of… ”) or a confidence interval (“estimate the … ”)?
Is the question dealing with quantitative data (e.g., means) or categorical data (e.g., proportions)?
How many (independent) samples do I have?
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8. Big Ideas Is the “difference” statistically significant?
p-value small?
How large is the difference? Estimation
Confidence intervals
Can we draw cause-and-effect conclusion?
Types of studies, Confounding variables
Random assignment?
Can we generalize to larger population?
Random sampling?
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9. Big Ideas Interpretations
What mean by “parameter”
What factors affect test statistic, p-value, confidence interval
What mean by “p-value”
Interpretation vs. evaluation
What mean by “confidence”
Study design issues (why…)
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10. What is “statistical inference”?
Need to be able to assess how much sample-to-sample variability there is, by chance alone (random sampling or random assignment, assuming the null hypothesis is true), to decide if an observed difference is “real”
Whether an observed sample result is surprising assuming some claim about the population (so estimate p-value)
How far we think our sample result might be from the population value (so estimate population value)
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11. Test of Significance Steps
Define parameter in words; State Ho and Ha hypotheses about parameter
Calculate/Interpret standardized statistic; Use technology to determine p-value
When allowed to use Theory-Based methods
Short-cut to running the simulation
Make conclusion: evaluate p-value, reject or fail to reject Ho, and answer the research question in context
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12. Confidence Interval Steps
Define parameter in words
Use technology to determine the interval
Theory-Based?
Two SD shortcut?
Interpret the interval
… I’m 90% confident that…
If asked, interpret “confidence” without using “confidence”
Maybe: Interpret the level
90% of intervals constructed this way
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13. Types of Interpretation Problems
Population vs. sample vs. null distributions (distribution of statistic)
Interpret probability, confidence interval, p-value, standardized statistic …
Including “what if”
Identify appropriate procedures
Scope of conclusions (generalizability, causation, significance, estimation)
Why?
confidence level,
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14. Common Mistakes Quantitative vs. categorical
Symbols: p vs. m (be able to define in words)
Technical conditions
z-procedures with proportions, t-procedures with means
Don’t combine: “mean proportion”
Stating hypotheses about statistic/nothing
H0: = .5
Sample size checks
vs. population check. What’s “normal”?
What changes when you increase the sample size…
Parameter vs. statistic, sample vs. null distribution
Stopping at “reject Ho” decision…
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15. Advice Work problems, don’t just read them!
Using only notes pages, calculator
Start with observation units, variable(s)
See if you can identify problem type
Think about what each step means
Banish the word IT (proof, data, group)
Add the phrase “of what” and “for what”
Learn from/seek clarification of grader comments
Labs, quizzes, investigations, etc.
Self-check, What went wrong, Self-tests
tables, computer,
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16. Quantitative data Penny ages Change Population Sample (n = 30)
Sampling distribution
Change
Population
Sample (n = 30)
Sampling distribution
Obs unit = sample
Variable = sample mean
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17. Example: Memory Study 1. Describe the sample
For our class, the average number of letters remembered was slightly higher for those receiving the JFK sequence (15.07 letters) than for those receiving the JFKC sequence (10.4 letters)
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18. Memory Study 1. Describe the sample
For our class, the average number of letters remembered was slightly higher for those receiving the JFK sequence (15.07 letters) than for those receiving the JFKC sequence (10.4 letters)
Multiples of three for JFK
Lots of overlap
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19. Memory Study 2. State hypotheses
Let mJFK represent the long-run mean number of letters that would be remembered with that sequence in population (similar for mJFKC)
H0: mJFK - mJFKC = 0 vs. Ha: mJFK - mJFKC > 0
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20. Memory Study
2. p-value Theory-based: sample sizes are not that large, sample distributions are not that normal looking so is questionable Could use simulation-based approach instead
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21. Finding p-value
Theory-based approach
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22. Confidence interval
(2.694) = (-.718, 10.06)
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23. Interpreting confidence interval
I’m 95% confident that the average JFK score is up to letters more than the average JFKC score but could also be .78 letters smaller.
I’m 90% confident that in the population, the JFK average score is .157 to 9.17 larger than the average JFKC score
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24. Summarizing conclusions
We have some evidence, but not super strong, that the JFK sequence leads to higher scores on average (theory-based p-value < .05)
We can say “leads to” because this was a randomized, double-blind experiment (independent evaluator of results)
In fact, we are 90% confident the average score is .16 to 9.2 more letters
But we are cautious in the population we generalize to as this was a convenience sample
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25. Factors
Original data
New data
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26. Factors
Original data
New data
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27. Factors
Original data
New data
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28. Questions? Office hours Optional review session Monday 11-12
Tuesday 2-3 in A
Optional review session
Wednesday 6pm (Library-111B)
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