Biostatistics Unit 5 – Samples. Sampling distributions Sampling distributions are important in the understanding of statistical inference. Probability.

Slides:



Advertisements
Similar presentations
Biostatistics Unit 5 Samples Needs to be completed. 12/24/13.
Advertisements

CmpE 104 SOFTWARE STATISTICAL TOOLS & METHODS MEASURING & ESTIMATING SOFTWARE SIZE AND RESOURCE & SCHEDULE ESTIMATING.
Chapter 10: Sampling and Sampling Distributions
Biostatistics Unit 4 Probability.
CHAPTER 8: Sampling Distributions
Biostatistics Unit 4 - Probability.
Chapter 7 Introduction to Sampling Distributions
PPA 415 – Research Methods in Public Administration Lecture 5 – Normal Curve, Sampling, and Estimation.
12.3 – Measures of Dispersion
Chapter 11: Random Sampling and Sampling Distributions
5.4 The Central Limit Theorem Statistics Mrs. Spitz Fall 2008.
Sample Distribution Models for Means and Proportions
Normal and Sampling Distributions A normal distribution is uniquely determined by its mean, , and variance,  2 The random variable Z = (X-  /  is.
Review of normal distribution. Exercise Solution.
Estimating a Population Mean
HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chapter 8 Continuous.
McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 7 Sampling Distributions.
Sampling Distributions
Albert Morlan Caitrin Carroll Savannah Andrews Richard Saney.
Population All members of a set which have a given characteristic. Population Data Data associated with a certain population. Population Parameter A measure.
16-1 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Chapter 16 The.
Estimation in Sampling!? Chapter 7 – Statistical Problem Solving in Geography.
1 Sampling Distributions Lecture 9. 2 Background  We want to learn about the feature of a population (parameter)  In many situations, it is impossible.
Lesson 9 - R Sampling Distributions. Objectives Define a sampling distribution Contrast bias and variability Describe the sampling distribution of a sample.
University of Ottawa - Bio 4118 – Applied Biostatistics © Antoine Morin and Scott Findlay 08/10/ :23 PM 1 Some basic statistical concepts, statistics.
Slide 1 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 1 n Learning Objectives –Identify.
Sampling Distributions Chapter 7. The Concept of a Sampling Distribution Repeated samples of the same size are selected from the same population. Repeated.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith.
Central Limit Theorem with Means
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 7.3.
McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 7 Sampling Distributions.
Sampling Distribution and the Central Limit Theorem.
Section 5.4 Sampling Distributions and the Central Limit Theorem Larson/Farber 4th ed.
Chapter 7 Sampling and Sampling Distributions ©. Simple Random Sample simple random sample Suppose that we want to select a sample of n objects from a.
1 Chapter 7 Sampling Distributions. 2 Chapter Outline  Selecting A Sample  Point Estimation  Introduction to Sampling Distributions  Sampling Distribution.
8 Sampling Distribution of the Mean Chapter8 p Sampling Distributions Population mean and standard deviation,  and   unknown Maximal Likelihood.
© Copyright McGraw-Hill 2000
Confidence Interval Estimation For statistical inference in decision making:
Copyright © 2009 Cengage Learning 9.1 Chapter 9 Sampling Distributions ( 표본분포 )‏
Chapter 9 Sampling Distributions Sir Naseer Shahzada.
Chapter 5 Sampling Distributions. The Concept of Sampling Distributions Parameter – numerical descriptive measure of a population. It is usually unknown.
Chapter 7: The Distribution of Sample Means. Frequency of Scores Scores Frequency.
Introduction to Inference Sampling Distributions.
From the population to the sample The sampling distribution FETP India.
Unit 6 Section : The Central Limit Theorem  Sampling Distribution – the probability distribution of a sample statistic that is formed when samples.
Sec 6.3 Bluman, Chapter Review: Find the z values; the graph is symmetrical. Bluman, Chapter 63.
+ Unit 5: Estimating with Confidence Section 8.3 Estimating a Population Mean.
Normal Probability Distributions 1 Larson/Farber 4th ed.
THE CENTRAL LIMIT THEOREM. Sampling Distribution of Sample Means Definition: A distribution obtained by using the means computed from random samples of.
Normal Probability Distributions
Chapter 6: Sampling Distributions
Sampling and Sampling Distributions
Sampling Distributions
The Central Limit Theorem
Introduction to Sampling Distributions
Chapter 6: Sampling Distributions
Chapter 7 Sampling Distributions.
Sampling Distributions and The Central Limit Theorem
Elementary Statistics: Picturing The World
Chapter 7 Sampling Distributions.
Econ 3790: Business and Economics Statistics
Warmup To check the accuracy of a scale, a weight is weighed repeatedly. The scale readings are normally distributed with a standard deviation of
Calculating Probabilities for Any Normal Variable
Chapter 7 Sampling Distributions.
Sampling Distributions
Chapter 7: The Distribution of Sample Means
PROBABILITY DISTRIBUTION
Chapter 7 Sampling Distributions.
Sampling Distributions and The Central Limit Theorem
Chapter 7 Sampling Distributions.
Presentation transcript:

Biostatistics Unit 5 – Samples

Sampling distributions Sampling distributions are important in the understanding of statistical inference. Probability distributions permit us to answer questions about sampling and they provide the foundation for statistical inference procedures.

Definition The sampling distribution of a statistic is the distribution of all possible values of the statistic, computed from samples of the same size randomly drawn from the same population. When sampling a discrete, finite population, a sampling distribution can be constructed. Note that this construction is difficult with a large population and impossible with an infinite population.

Construction of sampling distributions 1. From a population of size N, randomly draw all possible samples of size n. 2. Compute the statistic of interest for each sample. 3. Create a frequency distribution of the statistic.

Properties of sampling distributions We are interested in the mean, standard deviation and appearance of the graph (functional form) of a sampling distribution.

Types of sampling distributions We will study the following types of sampling distributions. A) Distribution of the sample mean B) Distribution of the difference between two meansDistribution of the sample meanDistribution of the difference between two means C) Distribution of the sample proportion D) Distribution of the difference between two proportionsDistribution of the sample proportionDistribution of the difference between two proportions

Sampling distribution of Given a finite population with mean (  ) and variance (   ). When sampling from a normally distributed population, it can be shown that the distribution of the sample mean will have the following properties.

Properties of the sampling distribution 1. The distribution of will be normal 2. The mean, of the distribution of the values of, will be the same as the mean of the population from which the samples were drawn; = . 3. The variance,, of the distribution of, will be equal to the variance of the population divided by the sample size; =.

Standard error The square root of the variance of the sampling distribution is called the standard error of the mean or the standard error.

Nonnormally distributed populations When the sampling is done from a nonnormally distributed population, the central limit theorem is used.

The central limit theorem Given a population of any nonnormal functional form with mean (  ) and variance (  2 ), the sampling distribution of, computed from samples of size n from this population will have mean, , and variance,  2 /n, and will be approximately normally distributed when the sample is large (30 or higher).

The central limit theorem Note that the standard deviation of the sampling distribution is used in calculations of z scores and is equal to

Example Given the information below, what is the probability that x is greater than 53? (1) Write the given information  = 50  = 16 n = 64 x = 53

Example (2) Sketch a normal curve

Example (3) Convert x to a z score

Example (4) Find the appropriate value(s) in the table A value of z = 1.5 gives an area of This is subtracted from 1 to give the probability P (z > 1.5) =.0668

Example (5) Complete the answer The probability that x is greater than 53 is.0668.

Distribution of the difference between two means It often becomes important to compare two population means. Knowledge of the sampling distribution of the difference between two means is useful in studies of this type. It is generally assumed that the two populations are normally distributed.

Sampling distribution of Plotting sample differences against frequency gives a normal distribution with mean equal to which is the difference between the two population means.

Variance The variance of the distribution of the sample differences is equal to Therefore, the standard error of the differences between two means would be equal to

Converting to a z score To convert to the standard normal distribution, we use the formula We find the z score by assuming that there is no difference between the population means.

Sampling from normal populations This procedure is valid even when Sampling from normal populations the population variances are different or when the sample sizes are different. Given two normally distributed populations with means, and, and variances, and, respectively. (continued)

Sampling from normal populations The sampling distribution of the difference,, between the means of independent samples of size n 1 and n 2 drawn from these populations is normally distributed with mean,, and variance,

Example In a study of annual family expenditures for general health care, two populations were surveyed with the following results: Population 1: n 1 = 40, = $346 Population 2: n 2 = 35, = $300

Example If the variances of the populations are = 2800 and = 3250, what is the probability of obtaining sample results as large as those shown if there is no difference in the means of the two populations?

Solution (1) Write the given information n 1 = 40, = $346, = 2800 n 2 = 35, = $300, = 3250

Solution (2) Sketch a normal curve

Solution (3) Find the z score

Solution (4) Find the appropriate value(s) in the table A value of z = 3.6 gives an area of This is subtracted from 1 to give the probability P (z > 3.6) =.0002

Solution (5) Complete the answer The probability that is as large as given is.0002.

Distribution of the sample proportion ( ) While statistics such as the sample mean are derived from measured variables, the sample proportion is derived from counts or frequency data.

Properties of the sample proportion Construction of the sampling distribution of the sample proportion is done in a manner similar to that of the mean and the difference between two means. When the sample size is large, the distribution of the sample proportion is approximately normally distributed because of the central limit theorem.

Mean and variance The mean of the distribution,, will be equal to the true population proportion, p, and the variance of the distribution,, will be equal to p(1-p)/n.

The z-score The z-score for the sample proportion is

Example In the mid seventies, according to a report by the National Center for Health Statistics, 19.4 percent of the adult U.S. male population was obese. What is the probability that in a simple random sample of size 150 from this population fewer than 15 percent will be obese?

Solution (1) Write the given information n = 150 p =.194 Find P( <.15)

Solution (2) Sketch a normal curve

Solution (3) Find the z score

Solution (4) Find the appropriate value(s) in the table A value of z = gives an area of.0869 which is the probability P (z < -1.36) =.0869

Solution (5) Complete the answer The probability that <.15 is.0869.

Distribution of the difference between two proportions This is for situations with two population proportions. We assess the probability associated with a difference in proportions computed from samples drawn from each of these populations. The appropriate distribution is the distribution of the difference between two sample proportions.

Sampling distribution of The sampling distribution of the difference between two sample proportions is constructed in a manner similar to the difference between two means. (continued)

Sampling distribution of Independent random samples of size n 1 and n 2 are drawn from two populations of dichotomous variables where the proportions of observations with the character of interest in the two populations are p 1 and p 2, respectively.

Mean and variance The distribution of the difference between two sample proportions,, is approximately normal. The mean is The variance is These are true when n 1 and n 2 are large.

The z score The z score for the difference between two proportions is given by the formula

Example In a certain area of a large city it is hypothesized that 40 percent of the houses are in a dilapidated condition. A random sample of 75 houses from this section and 90 houses from another section yielded difference,, of.09. If there is no difference between the two areas in the proportion of dilapidated houses, what is the probability of observing a difference this large or larger?

Solution (1) Write the given information n 1 = 75, p 1 =.40 n 2 = 90, p 2 =.40 =.09 Find P(.09)

Solution (2) Sketch a normal curve

Solution (3) Find the z score

Solution (4) Find the appropriate value(s) in the table A value of z = 1.17 gives an area of.8790 which is subtracted from 1 to give the probability P (z > 1.17) =.121

Solution (5) Complete the answer The probability of observing of.09 or greater is.121.

fin