Statistical Inference take what is known about the sample and infer that those traits apply to Exam 3 Bivariate Hypothesis testing Statistical Inference the population Parameters traits that can be quantified Averages Differences between groups Relationships among variables Sample Parameters Roman alphabet Population parameters Greek alphabet Samples to Populations Estimators of Population parameters get the Hat that thing on top Why do we take samples if we care about the population Feasibility Sometimes it would be impossible to document every occurrence of the phenomenon Cost Surveying all cases is often incredibly expensive Practicality We don t need to survey the full population Statistics why this class exists If don t right traits of the sample will accurately reflect traits of the population of interest The Batting average example if a guy is batting 333 he is likely to get a hit 1 in every 3 at bats 11 in every 33 at bats and so on Random Sample Simple Random Sample Ideal standard for a sample Every case in your population has an equal chance of being selected as part of the sample On average a simple random sample will reflect the population of interest These Simple Random samples are useful because of the Central Limit Theorem CLT Sampling Distribution We took a random sample for the population But it could have been any sample For any other random sample x is likely to be different the samples mean What if we did it many times We could GRAPH all of the observed sample means This graph of sample means is called the Sampling Distribution The Central Limit Theorem CLT CLT All sampling distributions follow a normal distribution in the limit If we took many samples the sample means would be normally distributed This DOES NOT say that the variable itself is normally distributed ONLY the Sampling Distribution is normally distributed An example Suppose I want to know what share of the vote the Democratic candidates receives in a Between 1946 and 2008 there were 11 720 contested congressional elections elections not Congressional elections counting special elections On average the Democratic candidate received 52 5 of the vote So 52 5 the true value This is The population parameter The true value Hypothetically speaking Imagine I did not have any data I could go out and collect the full population Or I could save time and effort and take a random sample How large a Sample Let s start with 10 that received votes in those 10 samples is x 57 38 If I take 10 more samples the sample mean is x 52 15 Getting us closer to our true mean of 52 5 the true value The CLT Suppose we took an infinite number of samples The mean of those samples would be Related concept The law of large numbers If I Randomly selects 10 Congressional elections I find that the mean of Democratic candidates If we repeat an experiment over and over again like rolling dice the sample mean would converge to the expected value Basically The more samples you take the average of those sample combined will come very close to representing the expected population mean the more samples you take the less difference between them As our sample size get larger our sample means are typically closer to the true mean What do we notice And fewer samples are way off What is happening For one our samples look more like the population Our sampling distribution is becoming Normal and more Leptokurtic Narrower These are good things As our sample size increases 1 Our observed sample means approach the populations mean 2 And the Standard error of the mean decreases So if you go from 10 to 100 to 1000 to 5000 samples the sampling distribution becomes more normal So what do we know We care about population parameters But we can usually only get estimates As sample size increases The expected value of our estimate equals the population value Sampling error decreases The sampling distribution takes a known form Normal distribution We are not going to be tested on the CLT formally It s on the power points if you saw it Uncertainty Our sample parameters are going to miss the mark But we need to have an idea of how far we are Standard error of the mean formula ox o n What does standard error tell us We can use a single sample to 1 estimate the population mean 2 Estimate how confident we are in that mean 3 See that as a sample size gets larger our confidence increases While standard error decreases Smaller Standard error Higher confidence Three random samples of numbers between 1 and 10 Size 10 100 1000 Mean Standard Dev Standard error Standard err total 4 20 5 12 5 07 2 62 3 06 3 09 2 62 10 3 06 100 3 09 1000 83 31 10 Why do we care Remember Sample means are normally distributed Also Standard error is the Standard deviation of the sampling distribution 95 of the sampling distribution lies plus or minus 1 96 standard deviations of the mean ASSUMING we have a Big sample Because of this we construct CONFIDENCE INTERVALS All this stuff is important try to understand it Confidence Interval of a mean To calculate the confidence interval of the mean we need to know 1 Sample mean Let s calculate at a 95 confidence interval 2 Sample Standard Deviation 3 Sample Size C I x 1 96 n An example from the chart Our Sample mean is 5 12 Our Standard error was 0 31 C I 5 12 1 96 0 31 1 96 0 31 61 rounded 5 12 0 61 5 73 There is a 2 5 chance the population mean is bigger than 5 73 5 12 0 61 4 51 There is a 2 5 chance the population mean is smaller than 4 51 To explain why its 2 5 since we want to calculate at a 95 confidence interval on both sides of the 1 96 interval which on the right would be positive and left would be negative sine there is only 5 left its split in half What if we had a smaller sample Recall our sample of 10 numbers Sample mean 4 20 Standard error 0 83 We can do the same thing 1 96 0 83 1 63 rounded The 95 confidence interval is 2 57 to 5 83 We re less confident of what the true mean is with a smaller sample Smaller the sample the larger the standard error J Bivariate Hypothesis testing What is Hypothesis testing 2 basic ideas to cover 1 Compare our sample estimate to hypothetical parameter 2 If Sample estimates of two groups are different We just can t look at raw estimates b c of uncertainty around our estimates Was the difference just random chance Research and the Null Hypothesis H1 The Research Hypothesis The difference we expect to see Between groups Between our estimate and hypothesized parameter H0 The Null Hypothesis Any difference
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