This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 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 zscore 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 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 When is unknown, well use SE ( X ) and the tdistribution 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 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 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....
View Full
Document
 Spring '10
 eee
 Standard Deviation, TDistribution

Click to edit the document details