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Unformatted text preview: 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 and standard deviation . 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 ( , 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 is unknown. We will estimate using the sample standard deviation s from a random sample, and estimate SD ( X ) = n by the standard error SE ( X ) = s n . For any particular X value, the corresponding z-score Z = X- SD ( X ) follows approxi- mately the standard normal distribution. When is unknown and SE ( X ) is used to estimate SD ( X ), the quantity X- SE ( X ) is no longer well modelled by the standard normal distribution. The sampling distribution of X- 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- SE ( X ) . The sampling distribution of T is called the Students 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 When is unknown, well use SE ( X ) and the t-distribution to construct confidence intervals for . A confidence interval of confidence level C for 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 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....
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