stat200ch7_fall10-3 - STAT 200 Chapter 7 Inference for...

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STAT 200 Chapter 7 Inference for Distributions Inferences based on the t-distribution (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 z-score 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 t -distribution . Properties of the t -distribution perfectly symmetric about the mean 0 unimodal, bell-shaped 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 t -distribution 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 “ t -score” such that the area to its right under the t -curve 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 one-sample t confidence interval . Assumptions and conditions for using the t-distribution 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 t -distribution is justified even if sample size is small. When the underlying distribution is skewed or non-normal, we will need a large sample for the t -distribution to work well. Hypothesis testing for the population mean μ – the one-sample t -test 1. Different forms of a hypothesis test on a population mean μ : H 0 : μ = μ o for some fixed μ o vs. (1) H a : μ = μ o (two-sided test) (2) H a : μ > μ o (one-sided, right-tailed test) (3) H a : μ < μ o (one-sided, left-tailed test)
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