STAT10x9.15.05TowardProbability

# STAT10x9.15.05TowardProbability - Toward Statistical...

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Toward Statistical Inference Thanks to Joe Chang in the Statistics Department for some lecture content Inference – use information about a sample to draw an inference about the population Example : A CNN/Gallup/USA Today poll of 1000 people reveal that 57% disapprove of Bush’s handling of the Katrina Hurricane disaster. We turn the fact that 57% of the sample have this opinion into an estimate that 57% of all people feel this way. Remember : Parameter – a fixed number that describes a population (i.e., μ = true population mean height). We don’t know this number (Gods only) Statistic – a number that describes a sample (i.e., x = sample mean height). We know this number, but the number can (and usually does) change from sample to sample . Use the statistic to estimate the unknown parameter! Example : MINITAB simulation - - - STAT 101 - 106 Introduction to Statistics 93

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Sampling Variability – If we repeated our sampling procedure ‘many’ times, the same way each time, how much would our statistics change from one sample to the next? Sampling Distribution of a statistic – the distribution of values of a sample statistic in all possible samples of the same size from a fixed population. Example : Let p be the true proportion of the population that approves of Bush (the PARAMETER). Suppose a TOTAL of FOUR people live in the U.S. (this is the population) . I.e., just this once, we know the entire population. Here are their opinions (known to Gods, who are letting us know just this once . . . ) We want to estimate p using a STATISTIC p ˆ , the sample proportion that approves of Bush. In this population, p =0.5 (the parameter, i.e. the true proportion of the population that approves of Bush). STAT 101 - 106 Introduction to Statistics 94 Individua l Attitude 1 Approve 2 Approve 3 Disapprov e 4 Disapprov e
NOW : Pretend we don’t know p =0.5, so we take a sample of size n =2. List all possible Simple Random Samples (SRS) of size 2 from this population, and record the sample proportion for each sample (the statistic , p ˆ ) POPULATION POSSIBLE SAMPLES In terms of probability, this is the sampling distribution for p ˆ for samples of size two : STAT 101 - 106 Introduction to Statistics 95 Individua l Attitude 1 Approve 2 Approve 3 Disapprov e 4 Disapprov e Individual s in SRS Attitude p ˆ 1 2 Dis Dis 0 1 3 App Dis 0.5 1 4 App Dis 0.5 2 3 App Dis 0.5 2 4 App Dis 0.5 3 4 App App 1 0 0.5 1 1/6 4/6 1/6

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The sampling distribution gives All possible values of p ˆ The proportion of times p ˆ takes on each of these values Now : what is the average value of the sample proportion, p ˆ ?. This is called the Mean of a sampling distribution. Mean of a sampling distribution = balancing point In this case, the Mean of sampling distribution of p ˆ = 0.5 This is also the value of the parameter p , the true proportion of the population that approves of Bush. If the mean of the sampling distribution of a statistic equals
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## This note was uploaded on 04/07/2008 for the course STAT 102 taught by Professor Jonathanreuning-schererdonaldgreen during the Fall '05 term at Yale.

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STAT10x9.15.05TowardProbability - Toward Statistical...

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