Lecture9standard

# Lecture9standard - Statistics 511 Statistical Methods Dr...

This preview shows pages 1–6. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

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: Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Lecture 11: Random Samples, Weak Law of Large Numbers and Central Limit Theorem Devore: Section 5.3-5.5 October, 2011 Page 1 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Definition of a Statistic • A statistic is any quantity whose value can be calculated from sample data. Prior to obtaining data, there is uncertainty as to what value of any particular statistic will result. • A statistic is a random variable denoted by an uppercase letter; a lowercase letter is used to represent the calculated or observed value of the statistic. October, 2011 Page 2 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 • Example Consider a sample of n = 3 cars of a particular type; their fuel efficiencies may be x 1 = 30 . 7 mpg, x 2 = 29 . 4 mpg, x 3 = 31 . 1 mpg. • It may also be x 1 = 28 . 8 mpg, x 2 = 30 . mpg and x 3 = 31 . 1 mpg • This implies that the value of the mean ¯ X is different in these cases. Clearly, ¯ X is a statistic. The first sample has the mean ¯ X 1 = 30 . 4 mpg and the second one has ¯ X 2 ≈ 30 mpg October, 2011 Page 3 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Statistic Examples • A sample mean ¯ X of the sample X 1 ,...,X n is a statistic; ¯ x is one of its possible values • The value of the sample mean from any particular sample can be regarded as a point estimate of the population μ . • Another example is the sample standard deviation S , while s is its computed value • Yet another example is the difference between the sample means for two different populations ¯ X- ¯ Y October, 2011 Page 4 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Sampling distribution • Each statistic is a random variable and, as such, has its own distribution • Consider two samples of size n = 2 ; if X 1 = X 2 = 0 , ¯ X = 0 with probability P ( X 1 = 0 ∩ X 2 = 0) • On the other hand, if X 1 = 1 but X 2 = 0 or X 1 = 0 and X 2 = 1 , we have ¯ X = 0 . 5 with probability P ( X 1 = 1 ∩ X 2 = 0) + P ( X 1 = 0 ∩ X 2 = 1) • This distribution is called the sampling distribution to emphasize its description of how the statistic varies in value across all possible sample October, 2011 Page 5 Statistics 511: Statistical Methods...
View Full Document

{[ snackBarMessage ]}

### Page1 / 21

Lecture9standard - Statistics 511 Statistical Methods Dr...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online