stat200ch7_winter10

# stat200ch7_winter10 - STAT 200 Chapter 7 Inference for...

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
This is the end of the preview. Sign up to access the rest of the document.

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 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....
View Full Document

## 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.

### Page1 / 12

stat200ch7_winter10 - STAT 200 Chapter 7 Inference for...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online