Unformatted text preview: 4/20/11 Midterm 1 Raw Scores Mul8ple Choice ques8ons (22 points) = 15.78 = 3.67 Short Answer (Graded out of 10 points) = 5.58 = 3.39 Overall: 30 points (MC + ( x SA) ) = 20.25 = 5.76 Midterm 1 Overall Results = 78.00% = 13.0% median: 78.8% Midterm 1 Announcements To view your exam, email your sec8on TA and he or she can bring it to sec8on To view your mul8ple choice answers, view the response report on WebCT Exam Materials Midterm 1 Response Report 1 4/20/11 Ques8ons numbers from Version A Refers to Ques8on 1 if you took Version A and Ques8on 10 if you took Version B 2 4/20/11 Put "E" as the answer Missed Ques8on 13 (as ordered on Version A) Missed Ques8on 13 (as ordered on Version A) Correct answer is C Put "E" as the answer Missed Ques8on 13 (as ordered on Version A) 3 4/20/11 This Week FR OM FO BE RE Parameters and Sta8s8cs Popula8on The Distribu8on of Sample Means The Logic of Hypothesis Tes8ng Sample ? ? A ESTIM TION X s STATISTICS (ESTIMATES) PARAMETERS FR BE OM FO RE Sampling Error The discrepancy (or amount of error) that exists between a sample sta8s8c and the corresponding popula8on parameter FR O BE M FO RE Popula8on All UCSD Students Two Possible Explana6ons: 1. There is no effect of caffeine on test scores and the 4point difference is due to sampling error 2. There is a real effect of caffeine on test performance No Caffeine Group Group 2 Sta6s6cs n = 50 82, 77 79, Average test score: 79 64, 81, 92, 74, 88, 91, 81, 79, 70... Caffeine Group n = 50 86, 76, 89, 91, 78, 89, 67, 54, 67, 88, 75, 90... Group 1 Sta6s6cs Average test score: 75 4 4/20/11 FR BE OM FO RE Single Sample Study Design 0 Sampling Error Popula8on All Individuals in the USA = 100 = 15 Popula8on All Individuals in the USA = 100 = 15 Treatment 5 4/20/11 Popula8on All Individuals in the USA = 100 = 15 Popula8on All Individuals in the USA = 100 = 15 Treatment Sample n = 25 Treatment Sample n = 25 106, 101, 98, 102, 100, 99, 111, 92, 108, 95, 110, 104... Popula8on All Individuals in the USA = 100 = 15 Popula8on All Individuals in the USA = 100 = 15 Treatment Sample n = 25 106, 101, 98, 102, 100, 99, 111, 92, 108, 95, 110, 104... Treatment Sample n = 25 106, 101, 98, 102, 100, 99, 111, 92, 108, 95, 110, 104... Sample Sta6s6cs X = 106 s = 15.2 Sample Sta6s6cs X = 106 s = 15.2 6 4/20/11 0 Sampling Error 0 Sampling Error Distribu8on of Sampling Error Distribu8on of Sampling Error 6 6 0 Sampling Error 0 Sampling Error Distribu8on of Sampling Error Distribu8on of Sampling Error 7 4/20/11 106 106 100 Popula8on Mean 100 Popula8on Mean Distribu8on of Sample Means Distribu8on of Sample Means The Distribu8on of Sample Means The Distribu8on of Sample Means The collec8on of sample means for all the possible random samples of a par8cular size (n) 8 4/20/11 Sampling Distribu8on A distribu8on of sta6s6cs obtained by compu8ng a sta8s8c for every possible random sample of size n from a popula8on The Distribu8on of Sample Means Sampling Distribu8on of X Sampling Distribu8on of M Construc8ng a real Sampling Distribu8on 1. 2. 3. 4. Choose a SMALL popula8on Enumerate every possible sample of size n Calculate a sta8s8c for each sample of size n Plot of histogram of the sta6s6cs calculated Construc8ng a real Sampling Distribu8on for a small popula6on 9 4/20/11 10 4/20/11 11 4/20/11 12 4/20/11 Check Your Understanding What is the probability we will get a sample with a mean of 2? A B C D E 3/16 5/16 1/16 1/4 1/8 Check Your Understanding What is the probability we will get a sample with a mean of 2? A B C D E 3/16 5/16 1/16 1/4 1/8 13 4/20/11 Check Your Understanding What is the probability we will get a sample with a mean greater than 6? A B C D E 3/16 5/16 1/16 1/4 1/8 Check Your Understanding What is the probability we will get a sample with a mean greater than 6? A B C D E 3/16 5/16 1/16 1/4 1/8 Construc8ng a real Sampling Distribu8on for a big popula6on 14 4/20/11 Construc8ng a real Sampling Distribu8on 1. 2. 3. 4. Choose a popula8on Enumerate every possible sample of size n Calculate a sta8s8c for each sample of size n Plot of histogram of the sta6s6cs calculated n! nP ! k (n " k)! 270! 270 P ! 2 (270 " 2)! 270! 270 P ! 2 (270 " 2)!
72630 different samples of size 2 15 4/20/11 270! 270 P ! 10 (270 " 10)! 270! 270 P ! 10 (270 " 10)!
1,739,366,431,045,825,933,612,800 different samples of size 10 Construc8ng a real Sampling Distribu8on 1. 2. 3. 4. 5. Choose a popula8on Take a random sample of size n Calculate the mean for that par6cular sample Record the mean to a histogram Go back to #2 and repeat 5000 8mes Popula8on Each Sample _ Sampling Distribu8on of X 16 4/20/11 = 100 = 15 = 100 = 15 Random Sample = 100 = 15 = 100 = 15 X X 17 4/20/11 = 100 = 15 = 100 = 15 Random Sample = 100 = 15 = 100 = 15 X X 18 4/20/11 = 100 = 15 Popula8on Each Sample = 100 = 15 _ Sampling Distribu8on of X Video of n=2 n = 2 (each random sample has two people) = 100 = 15 19 4/20/11 Video of n=4 n = 4 (each random sample has four people) = 100 = 15 n=2 vs n=4 5000 sample means for samples of size n = 2 5000 sample means for samples of size n = 4 20 4/20/11 Video of n=20 n = 20 (each random sample has twenty people) = 100 = 15 n=4 vs n=20 5000 sample means for samples of size n = 4 n = 60 (each random sample has sixty people) 5000 sample means for samples of size n = 20 21 4/20/11 Video of n=60 = 100 = 15 n=20 vs n=60 5000 sample means for samples of size n = 20 5000 sample means for samples of size n = 60 Video of n=120 n = 120 = 100 = 15 (each random sample has one hundred and twenty people) 22 4/20/11 n=60 vs n=120 5000 sample means for samples of size n = 60 5000 sample means for samples of size n = 120 = 100 = 15 Check Your Understanding n = 2 n = 2 n = 4 80 100 120 Roughly, what is the probability we will get a sample with a mean greater than 120? n = 20 n = 60 n = 120 A B C D E 0.00 0.05 0.20 0.30 0.50 23 4/20/11 Check Your Understanding n = 2 Check Your Understanding n = 20 Roughly, what is the probability we will get a sample with a mean greater than 120? 80 100 Roughly, what is the probability we will get a sample with a mean greater than 120? 80 100 120 120 A B C D E 0.00 0.05 0.20 0.30 0.50 A B C D E 0.00 0.05 0.20 0.30 0.50 Check Your Understanding n = 20 Characteris8cs of the Sampling Distribu8on of X Sampling distribu8on has a mean = As sample size (n) increases, the variability of the sampling distribu8on decreases The sampling distribu8on is normally distributed Roughly, what is the probability we will get a sample with a mean greater than 120? 80 100 120 A B C D E 0.00 0.05 0.20 0.30 0.50 24 4/20/11 Characteris8cs of the Sampling Distribu8on of X Sampling distribu8on has a mean = As sample size (n) increases, the variability of the sampling distribu8on decreases The sampling distribu8on is normally distributed Always?? Construc8ng a real Sampling Distribu8on for a big popula6on that is not normally distributed Construc8ng a real Sampling Distribu8on 1. 2. 3. 4. 5. Choose a popula8on (beta distribu8on) Take a random sample of size n Calculate the mean for that par6cular sample Record the mean to a histogram Go back to #2 and repeat 5000 8mes Popula8on Each Sample = 0.33 = 0.30 _ Sampling Distribu8on of X 25 4/20/11 = 0.33 = 0.30 = 0.33 = 0.30 Random Sample = 0.33 = 0.30 = 0.33 = 0.30 Random Sample X X 26 4/20/11 = 0.33 = 0.30 Popula8on Each Sample = 0.33 = 0.30 _ Sampling Distribu8on of X Video of n=2 n = 2 = 0.33 = 0.30 (each random sample has two people) 27 4/20/11 Video of n=5 n = 5 = 0.33 = 0.30 (each random sample has five people) n=2 vs n=5 5000 sample means for samples of size n = 2 (each random sample has twenty people) 5000 sample means for samples of size n = 5 n = 20 28 4/20/11 Video of n=20 = 0.33 = 0.30 n=5 vs n=20 5000 sample means for samples of size n = 5 5000 sample means for samples of size n = 20 Video of n=60 n = 60 = 0.33 = 0.30 (each random sample has sixty people) 29 4/20/11 n=20 vs n=60 5000 sample means for samples of size n = 20 (each random sample has sixty people) 5000 sample means for samples of size n = 60 n = 120 Video of n=120 = 0.33 = 0.30 n=60 vs n=120 5000 sample means for samples of size n = 60 5000 sample means for samples of size n = 120 30 4/20/11 = 0.33 = 0.30 Sampling Distribu8on of the Mean n = 2 Sampling Distribu8on of the Mean n = 5 Sampling Distribu8on of the Mean n = 20 Sampling Distribu8on of the Mean n = 60 Sampling Distribu8on of the Mean n = 120 Characteris8cs of the Sampling Distribu8on of X Sampling distribu8on has a mean = As sample size (n) increases, the variability of the sampling distribu8on decreases The sampling distribu8on is normally distributed when popula6on is normal or samples are large The Central Limit Theorem 31 4/20/11 "I know of scarcely anything so apt to impress the imagina8on as the wonderful form of cosmic order expressed by the [central limit theorem]. The theorem would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete selfeffacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chao8c elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beau8ful form of regularity proves to have been latent all along." Sir Francis Galton, 1889 The Central Limit Theorem For any popula8on with mean and standard devia8on , the distribu8on of sample means for sample size n will have a ! mean , and standard devia8on of n , and will approach a normal distribu8on as n approaches infinity Characteris8cs of the Sampling Distribu8on of X Sampling distribu8on has a mean = As sample size (n) increases, the variability of the sampling distribu8on decreases ! n The sampling distribu8on is normally distributed when popula6on is normal or samples are large, n > ~30 32 4/20/11 For Next Time Read: Chapter 7 (review) Chapter 8 Do Review Midterm Key 33 ...
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 Winter '09
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 Probability theory, Harshad number

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