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Unformatted text preview: Confindence Intervals and Probability Testing PO7001: Quantitative Methods I Kenneth Benoit 28 October 2009 Using probability distributions to assess sample likelihoods I Recall that using the and from a normal curve, we can then assess the probability of finding specific scores along this distribution (as we did last week) I The question: If we assume equals some specific value, then how likely was it to have drawn a given sample mean X? I A variation would be: If we draw two sample means, what is the probability that they have identical population means? Confidence intervals I Recall that a confidence interval is the range of values within which our population mean is most likely to fall I Confidence intervals are expressed as a percentage of confidence: If independent samples are taken repeatedly from the same population, and a confidence interval calculated for each sample, then a certain percentage (confidence level) of the intervals will include the unknown population parameter I Confidence intervals can be: I parametric: based on either assumed probability distributions I nonparametric: obtained through simulation (e.g. bootstrapping) Confidence intervals for proportions I Often we may seek to estimate a population proportion on the basis of a random sample: e.g. proportion to vote Yes on the Lisbon Treaty referendum I We use a version of the standard error known as the standard error of the proportion : s p = r p (1 p ) N where I s p is the standard error of the proportion I p is the sample proportion I N is the total number of cases in the sample I Because proportions have only a single parameter of which p is an estimate, we can use the normal distribution and not t I So: 95% CI for proportions = p 1 . 96 s p Confidence intervals for proportions: example Say, we want to look at the voter support for Vladimir Putin. 66% of 1600 respondents in a survey say they will vote for Putin. SE ( p ) = p p (1 p ) / n = p ( . 66 . 34) / 1600 = . 012 CI 95% = [ p 2 SE ( p ); p + 2 SE ( p )] = [ . 637; . 683] Confidence intervals for proportions: example 0.62 0.64 0.66 0.68 0.70 5 10 15 20 25 30 35 Support Putin Density Figure: Confidence interval of support for Putin Confidence intervals for proportions: example qnorm(c(.025, .975), .66, sqrt(.66 * .34 / 1600)) prop.test(.66 * 1600, 1600) Hypothesis testing I Hypothesis testing refers to a method of making statistical decisions using experimental data: a result is called statistically significant if it is unlikely to have occurred by chance I The null hypothesis ( H ) is the proposition that we will try to reject. It usually states the assumption that there is no effect or difference (depending on the research question) I The null hypothesis is stated as the opposite of what we would like to prove (more shortly) I The alternative hypothesis ( H 1 or H a ) states the research hypothesis which we will test assuming the null hypothesis . Steps in hypothesis testing...
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 Spring '08
 KARAALI

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