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Unformatted text preview: Introduction to Inference Estimating with Confidence Section 6.1 2009 W.H. Freeman and Company Overview of Inference ! Methods for drawing conclusions about a population from sample data are called statistical inference. ! Methods " Confidence Intervals- estimating a value of a population parameter " Tests of significance- assess evidence for a claim about a population ! Inference is appropriate when data are produced by either " a random sample or " a randomized experiment Uncertainty and confidence Although the sample mean, , is a unique number for any particular sample, if you pick a different sample you will probably get a different sample mean. In fact, you could get many different values for the sample mean, and virtually none of them would actually equal the true population mean, . x But the sample distribution is narrower than the population distribution, by a factor of ! n . Thus, the estimates gained from our samples are almost always relatively close to the population parameter ! . n Sample means, n subjects Population, x individual subjects If the population is normally distributed N ( ! , " ), then the sampling distribution is N ( ! , " / ! n ), Red dot: mean value of individual sample 95% of all sample means will be within roughly 2 standard deviations (2* " / ! n ) of the population parameter . Because distances are symmetrical, this implies that the population parameter must be within roughly 2 standard deviations from the sample average , in 95% of all samples. This reasoning is the essence of statistical inference. The weight of a single egg of the brown variety is normally distributed N (65 g,5 g). Think of a carton of 12 brown eggs as an SRS of size 12. . You buy a carton of 12 white eggs instead. The box weighs 770 g. The average egg weight from that SRS is thus = 64.2 g. ! Assuming that the standard deviation of white egg weight is also 5 g, what can you infer about the mean ! of the white egg population? We are 95% confident that the interval 64.2g 2.88g contains the population mean ! population sample ! What is the distribution of the sample means ? Normal (mean , standard deviation " / ! n ) = N (65 g,1.44 g). ! Find the middle 95% of the sample means distribution. Roughly 2 standard deviations from the mean, or 65g 2.88g. Confidence interval The confidence interval is a range of values with an associated probability or confidence level C . The probability quantifies the chance that the interval contains the true population parameter. 4.2 is a 95% confidence interval for the population parameter . This equation says that in 95% of the cases, the actual value of will be within 4.2 units of the value of ....
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- Fall '08