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IPS6eCh06_1_2bb

IPS6eCh06_1_2bb - Introduction to Inference Estimating with...

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Objectives (IPS Chapter 6.1) Estimating with confidence Statistical confidence Confidence intervals Confidence interval for a population mean How confidence intervals behave Choosing the sample size Introduction to Inference Estimating with Confidence

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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
Statistical 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, μ . x

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But the sample distribution is narrower than the population distribution, by a factor of √ n . Thus, the estimates gained from our samples are always relatively close to the population parameter µ . n Sample means, n subjects μ n σ σ Population, x individual subjects x x If the population is normally distributed N ( µ , σ ), so will the sampling distribution N ( µ , σ /√ n ),
Red dot: mean value of individual sample 95% of all sample means will be within roughly 2 standard deviations (2* σ /√ n ) of the population parameter μ . Distances are symmetrical which implies that the population parameter μ must be within roughly 2 standard deviations from the sample average , in 95% of all samples. This reasoning is the essence of statistical inference. σ n x

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The weight of single eggs 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. Knowing that the standard deviation of egg weight is 5 g, what can you infer about the mean µ of the white egg population? There is a 95% chance that the population mean µ is roughly within ± 2 σ /√ n of , or 64.2 g ± 2.88 g. population sample What is the distribution of the sample means ? Normal (mean μ , standard deviation σ /√ 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. x x x
Confidence intervals 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 μ .

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IPS6eCh06_1_2bb - Introduction to Inference Estimating with...

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