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# 18 Sampling Mean Sd

## on Mar 23, 2010

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## 18 Sampling Mean SdPresentation Transcript

• Stat310 Sampling distributions Hadley Wickham Tuesday, 23 March 2010
• 1. About the test 2. Sampling distribution of the mean 3. Sampling distribution of the standard deviation Tuesday, 23 March 2010
• Test Next Tuesday. Covers bivariate random variables and inference up to Thursday. Same format as last time: 4 questions, 80 minutes. 2 sides of notes. Half applied and half theoretical. Hopefully a little easier than last time. Tuesday, 23 March 2010
• Test tips Work through the learning objectives online, looking them up in your notes if you’re not sure. Work through the practice problems. Go back over previous quizzes and homeworks and make sure you know how to answer each question. Tuesday, 23 March 2010
• Sampling distribution of the mean Tuesday, 23 March 2010
• Means X1, X2, ... are iid N(μ, σ2) n ￿ ¯ Sn Sn = Xi Xn = n 1 Then 2 σ ¯ n ∼ N(µ, ) X n Tuesday, 23 March 2010
• Means X1, X2, ... are iid N(μ, σ2) n ￿ ¯ Sn Sn = Xi Xn = n 1 Then 2 σ ¯ n ∼ N(µ, ) X n Tuesday, 23 March 2010
• Means X1, X2, ... are iid E(X) = μ, Var(X) = σ2 n ￿ ¯ Sn Sn = Xi Xn = n 1 Then 2 σ ¯ n ∼ N(µ, ) X ˙ n Tuesday, 23 March 2010
• Means X1, X2, ... are iid E(X) = μ, Var(X) = σ2 n ￿ ¯ Sn Sn = Xi Xn = n 1 Then 2 σ ¯ n ∼ N(µ, ) X ˙ n Tuesday, 23 March 2010
• Means X¯n − µ Zn = 2 √ σ / n Zn ∼ N(0, 1) ˙ Tuesday, 23 March 2010
• Your turn Back to the Lakers. Let Oi ~ Poisson(λ = 103.9) - their offensive score for a single game. What is the distribution of their average score for the entire season? (There are 82 games in a season) Tuesday, 23 March 2010
• Continuity correction When using the normal distribution to approximate a discrete distribution we need to make a small correction P(X = 1) = P(0.5 < Z < 1.5) P(X < 1) = P(Z < 0.5) P(X ≤ 1) = P(Z < 1.5) P(X > 1) = P(Z > 1.5) Tuesday, 23 March 2010
• Your turn What’s the probability the average score for the Lakers is less than 100? Tuesday, 23 March 2010
• Steps Write as probability statement. Transform each side to get to known distribution. Apply continuity correction, if necessary. Compute. Tuesday, 23 March 2010
• Multiplication X ~ Poisson(λ) Y = tX Then Y ~ Poisson(λt) Tuesday, 23 March 2010
• Exactly How could you use the Poisson distribution to calculate the exact probability that the average score is < 100? Tuesday, 23 March 2010
• Sampling distribution of the standard deviation Tuesday, 23 March 2010
• (n − 1)S 2 2 ∼ χ (n − 1) 2 σ ￿ ¯ 2 (Xi − X) If Xi ~ iid N(0, 1), S = 2 n−1 Tuesday, 23 March 2010
• Five fun facts about 2 χ Tuesday, 23 March 2010
• Proof (n − 1)S 2 2 ∼ χ (n − 1) 2 σ Tuesday, 23 March 2010
• Sampling distribution of mean if variance unknown Tuesday, 23 March 2010
• Your turn When we have to estimate the sd, what do you think happens to the distribution of our estimate of the mean? (Would it get more or less accurate?) What about as n gets bigger? Tuesday, 23 March 2010
• 0.3 df 1 dens 0.2 2 15 Inf 0.1 −3 −2 −1 0 1 2 3 x Tuesday, 23 March 2010
• t-distribution Xi ∼ Normal(µ, σ )2 ¯n − µ X ¯n − µ X √ ∼Z √ ∼ tn−1 σ/ n s/ n Parameter called degrees of freedom Tuesday, 23 March 2010
• 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. Tuesday, 23 March 2010
• 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 exams, I’ll give you a small table if necessary) Tuesday, 23 March 2010