Unit4: Inferencefornumericaldata. 2. Bootstrapping
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1 Announcements Unit4: Inferencefornumericaldata 2. Bootstrapping Sta Summer 2017 Duke University, Department of Statistical Science Project Description now posted on course page Project time in lab on Monday, July 24. Discuss with your team members, ask questions if you re stuck! Lab due on Monday PS 4 and PA 4 both due next Wednesday Prof. Burris Slides posted at 1 Readiness assessment Rotten horrors Individual: 15 minutes, using bubble sheets is a movie aggregator, where the audience is also able to review and score the movies. We want to estimate the average audience score of horror movies on RottenTomatoes.com. We start with a random sample of 20 horror movies. Team: 10 minutes, using scratch off sheets (1 per team) 2 3
2 Data First look 1 Patrick 52 2 Demon Seed 43 3 Tormented 34 4 Under the Bed 12 5 Phantasm IV: Oblivion 41 6 Fright Night Part House of 1000 Corpses 65 8 Creepshow The Forsaken All the Boys Love Mandy Lane Jason Lives: Friday the 13th Part VI Vampire's Kiss The Witches of Eastwick Yellowbrickroad Dying Breed Carrie Whoever Slew Auntie Roo? The Mangler Primal The Twilight Saga: New Moon 65 The histogram below shows the distribution of the audience scores of these movies (ranging from 0 to 100). The median score in the sample is Can we apply CLT based methods we have learned so far to construct a confidence interval for the median RottenTomatoes score of horror movies. Why or why not? audience scores 4 5 Bootstrapping Bootstrapping An alternative approach to constructing confidence intervals is bootstrapping. This term comes from the phrase pulling oneself up by one s bootstraps, which is a metaphor for accomplishing an impossible task without any outside help. In this case the impossible task is estimating a population parameter, and we ll accomplish it using data from only the given sample. Bootstrapping works as follows: (1) take a bootstrap sample - a random sample taken with replacement from the original sample, of the same size as the original sample (2) calculate the bootstrap statistic - a statistic such as mean, median, proportion, etc. computed on the bootstrap samples (3) repeat steps (1) and (2) many times to create a bootstrap distribution - a distribution of bootstrap statistics The XX% bootstrap confidence interval can be estimated by the cutoff values for the middle XX% of the bootstrap distribution, OR point estimate ± t SE boot 6 7
3 Bootstrap sample 1 Bootstrap sample 2 (1)Takeabootstrapsample: 1 Vampire's Kiss 48 2 Phantasm IV: Oblivion 41 3 House of 1000 Corpses 65 4 Dying Breed 27 5 Whoever Slew Auntie Roo? 56 6 The Forsaken 44 7 The Twilight Saga: New Moon 65 8 The Twilight Saga: New Moon 65 9 Whoever Slew Auntie Roo? The Twilight Saga: New Moon The Mangler Dying Breed Creepshow House of 1000 Corpses Whoever Slew Auntie Roo? Tormented Jason Lives: Friday the 13th Part VI Vampire's Kiss Primal The Witches of Eastwick 60 23, 27, 27, 29, 34, 41, 44, 46, 48, 48, 56, 56, 56, 57, 60, 65, 65, 65, 65, 65 median = ( ) / 2 = 52 (1)Takeanotherbootstrapsample: 1 Fright Night Part Carrie 73 3 The Forsaken 44 4 The Mangler 23 5 Primal 29 6 Patrick 52 7 Jason Lives: Friday the 13th Part VI 57 8 The Mangler 23 9 Vampire's Kiss All the Boys Love Mandy Lane The Twilight Saga: New Moon All the Boys Love Mandy Lane Yellowbrickroad Vampire's Kiss Tormented The Mangler Phantasm IV: Oblivion Patrick House of 1000 Corpses The Twilight Saga: New Moon 65 23, 23, 23, 28, 29, 34, 34, 34, 41, 42, 44, 48, 48, 52, 52, 57, 65, 65, 65, 73 median = ( ) / 2 = Bootstrap sample 3 Many more bootstrap samples (1)Takeanotherbootstrapsample: 1 Tormented 34 2 The Witches of Eastwick 60 3 The Witches of Eastwick 60 4 The Witches of Eastwick 60 5 The Mangler 23 6 The Witches of Eastwick 60 7 Patrick 52 8 Phantasm IV: Oblivion 41 9 Yellowbrickroad Jason Lives: Friday the 13th Part VI Yellowbrickroad Jason Lives: Friday the 13th Part VI Fright Night Part Primal Fright Night Part Whoever Slew Auntie Roo? Fright Night Part Fright Night Part Under the Bed Phantasm IV: Oblivion repeat 12, 23, 28, 28, 29, 34, 41, 41, 42, 42, 42, 42, 52, 56, 57, 57, 60, 60, 60, 60 median = ( ) / 2 =
4 Question The dot plot below is the bootstrap distribution of medians constructed using 100 simulations. What does each dot on the dot plot represent? Question The dot plot below shows the distribution of 100. Estimate the 90% bootstrap confidence interval for the median RT score of horror movies using the percentile method. (a) Score of a horror movie in the original sample (b) Score of a horror movie in the population (c) Median from one bootstrap sample from the original sample (d) Median from one sample from the population (a) (29, 58.5) (b) (34, 57) (c) (37.5, 52) (d) (40, 49.5) Bootstrap interval, standard error Bootstrap testing for a mean The dot plot below shows the distribution of 100. The median of the original sample is 43.5 and the bootstrap standard error is Estimate the 90% bootstrap confidence interval for the median RT score of horror movies using the standard error method. This is very similar to bootstrapping, i.e. we randomly sample with replacement from the sample, but this time we shift the bootstrap distribution to be centered at the null value. The p-value is then defined as the proportion of simulations that yield a sample statistic at least as favorable to the alternative hypothesis as the observed sample statistic
5 Bootstrap vs. sampling distributions Do these data provide convincing evidence that the median audience score of horror movies is greater than 40? Remember that the median of the original sample was Application exercise: 4.2 Bootstrap intervals See the course webpage for details. H 0 : median = 40 H A : median > 40 p-value: proportion of simulations where the simulated bootstrap sample median is at least as extreme as the one observed (43.5). 20 / 100 = Summary of main ideas 1. Bootstrapping = sampling with replacement from the observed sample 2. Bootstrap percentile intervals: middle XX% of the bootstrap distribution 3. Bootstrap SE intervals: point estimate ± ME 4. Bootstrap testing for a single numerical variable requires shifting the bootstrap distribution to be centered at the null value 18
Unit4: Inferencefornumericaldata. 2. Bootstrapping
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