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Unformatted text preview: STAT 200 Chapter 7 Inference for Distributions Inferences based on the tdistribution (Section 7.1) • Let X be a random variable that follows a certain distribution with mean μ and standard deviation σ . If we have a large enough sample size n , and other necessary conditions being satisfied, then the sampling distribution of the sample means X follows approximately N ( μ, σ √ n ). If the underlying distribution of X is normal, the sampling distribution is normal regardless of the sample size. • In most situations, the value of the population standard deviation σ is unknown. We will estimate σ using the sample standard deviation s from a random sample, and estimate SD ( X ) = σ √ n by the standard error SE ( X ) = s √ n . • For any particular X value, the corresponding zscore Z = X μ SD ( X ) follows approxi mately the standard normal distribution. When σ is unknown and SE ( X ) is used to estimate SD ( X ), the quantity X μ SE ( X ) is no longer well modelled by the standard normal distribution. The sampling distribution of X μ SE ( X ) (we obtain one x for each repeated sample of a fixed sample size n ) has thicker tails than the standard normal distribution. Also, the shape of the distribution changes with the sample size. We use “ T ” to denote this quantity, T = X μ SE ( X ) . The sampling distribution of T is called the Student’s tdistribution . • Properties of the tdistribution – perfectly symmetric about the mean 0 – unimodal, bellshaped – has one parameter – the degrees of freedom ( df ) which determines the shape of the distribution and is given by n 1 – has thicker tails when sample size is smaller – approaches the standard normal distribution as sample size increases Confidence intervals for the population mean μ • When σ is unknown, we’ll use SE ( X ) and the tdistribution to construct confidence intervals for μ . A confidence interval of confidence level C for μ is constructed using: 1 x ± t * n 1 s √ n where the critical point t * n 1 is the “ tscore” such that the area to its right under the tcurve with n 1 degrees of freedom is equal to 100% confidence level 2 (See Table D in your textbook). Such a confidence interval for μ is called the onesample t confidence interval . • Assumptions and conditions for using the tdistribution for statistical inferences – the sample is randomly drawn from the population – the sampled values are independent (sample size is a small fraction of the popu lation size) – when the underlying distribution of X is nearly normal, or unimodal and sym metric, using the tdistribution is justified even if sample size is small. When the underlying distribution is skewed or nonnormal, we will need a large sample for the tdistribution to work well....
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This note was uploaded on 10/21/2010 for the course STATISTICS Stat 200 taught by Professor Eee during the Spring '10 term at UBC.
 Spring '10
 eee
 Standard Deviation, TDistribution

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