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### 22 confidence

1. 1. Stat310 Conﬁdence intervals Hadley Wickham Thursday, 15 April 2010
2. 2. Quiz • Pick up quiz on your way in • Start at 1pm • Finish at 1:10pm • Closed book Thursday, 15 April 2010
3. 3. 1. Test extra credit 2. Inference roadmap 3. Steps for making a conﬁdence interval 4. One more sampling distribution (the t-distribution) Thursday, 15 April 2010
4. 4. I s rt til y p pa Test makeup l n la ee nn d er on ! e Homework graded out 10 4% of overall grade (= 20% of two tests) Will act as extra credit for the test. (i.e. there is no penalty you don’t do it) Due next Thursday. The extra 5% of the grade will be distributed across all other assessment. Thursday, 15 April 2010
5. 5. What we want to do Given data: • Estimate true value of parameter (last week) • Quantify uncertainty of estimate (today) • Test whether true value is a certain value (Thursday) Thursday, 15 April 2010
6. 6. Tools • Construct an estimator • Method of moments • Maximum likelihood • Work out its distribution • Sampling distribution of mean • Sampling distribution of variance • General properties of ML (not in this course) Thursday, 15 April 2010
7. 7. Set up I repeated an experiment deﬁned by Poisson(λ) 10 times, and recorded the following results: 6 11 10 6 12 7 8 5 7 10 The MLE of λ is 8.2, and its standard deviation is 0.90. What is the distribution of the estimate? (Remember that it’s a mean) Can you construct an interval that will contain λ 95% of the time? Thursday, 15 April 2010
8. 8. Steps 1. Identify distribution that connects estimator and true value (4 choices). 2. Form conﬁdence interval for known (sampling) distribution, and work out bounds. 3. Back transform. 4. Write as interval. 5. Plug in sample estimates (actual numbers). Thursday, 15 April 2010
9. 9. Your turn Work through the steps on the handout. Thursday, 15 April 2010
10. 10. Conﬁdence interval A conﬁdence interval is a simple numerical summary of the uncertainty of an estimate. A 95% conﬁdence interval will contain the true value 95% of the time. An additional constraint is that we want the conﬁdence interval to be a short as possible. Thursday, 15 April 2010
11. 11. Each line = 95% conﬁdence interval from one experiment 12 11 10 9 8 50 100 150 200 expt Thursday, 15 April 2010
12. 12. Horizontal line = true value 12 11 10 9 8 50 100 150 200 expt Thursday, 15 April 2010
13. 13. Red intervals don’t include true value 12 11 10 9 8 50 100 150 200 expt There are 13 red lines and 200 experiments. Is this an ok interval? Thursday, 15 April 2010
14. 14. Your turn What’s wrong with a statement like this: P(2 < μ < 6) = 0.95 ? Thursday, 15 April 2010
15. 15. Steps Identify distribution that connects estimator and true value. Form conﬁdence interval for known (sampling) distribution. Write as probability statement. Back transform. Write as interval. Thursday, 15 April 2010
16. 16. Xi iid, and n large: ¯n − µ . X √ ∼Z σ/ n Thursday, 15 April 2010
17. 17. iid 2 Xi ∼ Normal(µ, σ ) (n − 1)S2 2 ∼ χ (n − 1) 2 σ X ¯n − µ √ ∼Z σ/ n X ¯n − µ √ ∼ tn−1 s/ n Thursday, 15 April 2010
18. 18. 0.3 df 1 dens 0.2 2 15 Inf 0.1 −3 −2 −1 0 1 2 3 x Thursday, 15 April 2010
19. 19. Properties of the t-dist Heavier tails compared to the normal distribution. lim tn = Z n→∞ Practically, if n > 30, the t distribution is practically equivalent to the normal. Thursday, 15 April 2010
20. 20. t-tables Basically the same as the standard normal. But one table for each value of degrees of freedom. Easiest to use calculator or computer: http://www.stat.tamu.edu/~west/applets/ tdemo.html (For homework, use this applet, for ﬁnal, I’ll give you a small table, if necessary) Thursday, 15 April 2010
21. 21. Thursday, 15 April 2010
22. 22. Your turn We perform the experiment an experiment to measure the speed of sound and repeat it 10 times: 340 333 334 332 333 336 350 348 331 344 (mean: 338, sd: 7.01) Assuming Xi ~ Normal(μ, σ2), what is an estimate of the speed of sound? What is the error (sd) of this estimate? Give an interval that we’re 95% certain the true speed of sound lies in. Thursday, 15 April 2010
23. 23. Example 340 333 334 332 333 336 350 348 331 344 (mean: 338, sd: 7.01) If not known: (333, 342) (2.23) Thursday, 15 April 2010
24. 24. Steps Identify distribution that connects estimator and true value. Form conﬁdence interval for known (sampling) distribution. Write as probability statement. Back transform. Write as interval. Thursday, 15 April 2010
25. 25. Steps Want P(a < Q < b) = 1 - α, and b - a to be as small as possible. If Q is symmetric, P(-a < Q < a) = 1 - α. So a = F(α/2), and there is no interval smaller. If Q isn’t symmetric, pick a = F(α/2), b = F(1 - α/2), but there might be a shorter interval. Thursday, 15 April 2010
26. 26. Example We want a 90% conﬁdence interval, then two possible ends for the interval are F(0.05) and F(0.95) Thursday, 15 April 2010
27. 27. Reading Read the rest of chapter 6. Everything else is just examples of the general method we learned today. Thursday, 15 April 2010