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MIT 9 07 - Lecture Slides

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Last few slides from last time…Binomial distribution • Distribution of number of successes in n independent trials. • Probability of success on any given trial = p • Probability of failure on any given trial = q = 1-pMean and variance of a binomial random variable • The mean number of successes in a binomial experiment is given by: – µ = np – n is the number of trials, p is the probability of success • The variance is given by – σ2 = npq –q = 1-p25 coin flips • What is the probability that the number of heads is ≤ 14? • We can calculate from the binomial formula that p(x≤14) is .7878 (note this is not an approximation)Normal Approximation • Using the normal approximation with µ = np = (25)(.5) = 12.5 and σ = sqrt(npq) = sqrt((25)(.5)(.5)) = 2.5 we get • p(x≤14) = p(z ≤ (14-12.5)/2.5)) = p(z ≤.6) = .7257 • .7878 vs. .7257 -- not great!! • Need a better approximation...Normal Approximation of Binomial Distribution 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 p(x) 0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425 Number of SuccessesContinuity Correction• Notice that the bars are centered on the numbers• This means that p(x≤14) is actually the area under the bars less than x=14.5• We need to account for the extra 0.5•P(x≤14.5) = p(z≤.8) = .7881 -- a much better approximation!9 1011121314151617Number of SuccessesSampling theory 9.07 2/24/2004Goal for rest of today • Parameters are characteristics of populations. – E.G. mean, variance • We’ve also looked at statistics* of a sample. – E.G. sample mean, sample variance • How good are our statistics at estimating the parameters of the underlying population? *Statistic = a function of a sample. There are a lot of possible statistics, but some are more useful than others.Experimental design issues in sampling • Suppose we want to try to predict the results of an election by taking a survey. – Don’t want to ask everyone – that would be prohibitive! – What percent of voters will vote for the Republican candidate for president? – We ask 1000 eligible voters who they will vote for. – In the next lecture, we will talk about estimating the population parameter (the % of voters who prefer the Republican) from the % of the survey respondents who say they favor him. – But our statistics are only as good as our sampling technique – how do we pick those 1000 people for the survey?Simple random sample • If the procedure for selecting n objects out of a large population of objects is such that all possible samples of n objects are equally likely, then we call the procedure a simple random sample. • This is the gold standard for sampling. – Unbiased: each unit has the same probability of being chosen. – Independent: selection of one unit has no influence on selection of other units.In theory, how to get a simple random sample • Get a list of every unit in the population. • Randomly pick n objects using, e.g. a random number generator. • Or, put a card for each unit in a drum, and pull out n cards (without looking) • This may be prohibitive…Opportunity sampling • Take the first n people who volunteer • The population of people who volunteer may be quite different from the general population. – Shere Hite: 100000 questionaires for book “Women & Love” left lying about in women’s organizations. Women returned the questionaires if they wanted to. • Came under fire because women in women’s organizations who volunteer to fill out the survey may have very different attitudes toward sex and love than the general population of women.Opportunity sampling • Nonetheless, in human studies, we do a lot of opportunity sampling in BCS. • We’re hoping there’s not much difference between people who sign up for a cog sci experiment and the general population. • In many cases this is probably not a bad assumption, but beware! • Volunteering aside, many of the subjects are MIT students…Simple random sample • The methods described in this class apply to a simple random sample. If you don’t have one, the methods need to be modified. • Randomized design is key to trusting your statistics.Sampling theory • A statistic can be computed from a sample, and used to estimate a parameter (of the population). • A statistic is what the investigator knows. • A parameter is what the investigator wants to know. • When estimating a parameter, one major issue is accuracy: how close is the estimate going to be. Freedman et al, StatisticsHow good is our estimator? • As an example, consider estimating the population mean, µ, with the mean of N samples, m. • Bias: – If E(m) = µ, the estimator is unbiased. – If E(m) = µ’, the bias is µ’- µ – All else being equal, you’d prefer that your estimate of the mean would, on average, equal the population mean, instead of, e.g., being smaller than µ, on average.How good is our estimator? • Consistency: – If the estimator gets better as we apply it to a largersample, then the estimate is consistent. • Relative efficiency: – Just as estimators have a mean, they also have a variance. – If G and H are both unbiased estimators of µ, then the more efficient estimator is the one with the smaller variance. – Efficient estimators are nice because they give you less chance error in your estimation of the population parameter.How good is our estimator? • Sufficiency – If an estimator G contains all of the information in the data about parameter µ, then G is a sufficient estimator, or sufficient statistic. That is, if G is a sufficient statistic, we can’t get a better estimate of µ by considering some aspect of the data not already accounted for in G.What do we need, to decide how good our estimator is? • Well, we need to know its mean and variance, so we can judge its bias and efficiency. • It’d also be nice to know, more generally, what is the distribution of values we expect to get out ofour estimator, for a given set of population parameters. • Example: what is the distribution of the sample mean, given that the population has mean µ and variance σ2 ? – This is called the sampling distribution of the mean. – You already estimated it on one of your homeworks.“Sampling distribution” of the mean • On the last homework, you generated 100000 examples of 5 samples from a normal


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MIT 9 07 - Lecture Slides

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