Probability and Samples: The Distribution of Sample Means Chapter 7

Chapter Overview Samples and Sampling Error The Distribution of Sample Means Probability and the Distribution of Sample Means Computations

Q? What is the purpose of obtaining a sample? A. To provide a description of a population

What happens when the sample mean differs from population mean? Sampling Error: The discrepancy, or amount of error, between a sample statistic and its corresponding population parameter. 2 separate samples from the same population will probably differ. different individual different scores different sample means How can you tell if the sample is giving the best description of the population? In order to determine how well a sample will describe its population, there is a systematic, orderly set of predictable patterns to help predict the characteristics of a sample. This is called the distribution of sample means These questions can be answered once we establish a set of rules that related samples to populations.

Predicting the characteristics of a sample Distribution of Sample Means: the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population N=1 P=1/100 Do not confuse this with the distribution of single scores.

Distribution of sample means are statistics, not single scores. Sampling distribution: a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.

Example 7.1

Let’s construct a distribution of sample means What do we need to know Population parameters (scores) 2,4,6,8 Specify an (n) Examine all possible samples

Figure 7.2 What is the probability of obtaining a score greater than 7? P 1/16

Characteristics of sample means Sample means tend to pile up around the population mean The distribution of sample means is approximately normal in shape. The distribution of sample means can be used to answer probability questions about sample means

What do we use when we have a large n and do not want to calculate all of the possible samples ?

Central Limit Theorem CLT: For any population with mean of  and a standard deviation , the distribution of sample means for sample size n will approach a normal distribution with a mean of  and a standard deviation of /n (square root of n) as n approaches infinity. Useful b/c it is impossible to obtain all of the samples means. This theorem can be applied to any population, no matter what the shape, or mean, or sd. The distribution of sample means approaches a normal distribution

CLT: Facts Describes the distribution of two sample of sample means for any population, no matter what shape, mean, or standard deviation. The distribution of sample means “approaches” a normal distribution by the time the size reaches n= 30.

Central Limit Theorem Cont’d Distribution of sample means tends to be a normal distribution particularly if one of the following is true: The population from which the sample is drawn is normal. The number of scores (n) in each sample is relatively large (n>30)

Expected value of X Sample means should be close to the population mean aka the expected value of x Expected value of X: the mean of the distribution of sample means will be equal to  (the population mean) Average number of sample means should equal the population mean. The sample mean is an unbiassed estimate of the population mean

Standard Error of X Notation: x = standard distance between x and  The standard deviation of the distribution of sample means. Measures the standard amount of difference one should expect between X and  simply due to chance The single standard deviation of a sample mean from the population mean The standard error of the sample mean is extremely valuable because it specifies how well our sample estimates a population mean.

Magnitude of the Standard error is determined by The size of the sample The standard deviation of the population from which the sample is selected Law of large numbers: the > n, the more probable the sample mean will be close to the population mean. As sample size decreases, the standard error increases.

Learning Check pg 151 A population of scores is normal with =80 and =20 Describe the distribution of sample means for samples of size n=16 selected from this population. (Describe shape, central tendency, and variability, for the distribution) How would the distribution of sample means be changed if the sample size were n=100 instead of n=16. 1) The distribution of sample means will be normal with an expected mean of 20 and a standard error of 20 divided by the square root of 16 = 5

2) As sample size increases, the value of the standard error also increases? (True or False) 3)Under what circumstances will the distribution of sample means be a normal shaped distribution? 3) Sample size if more than 30 or if the population is normal

Learning Check 7.2 pg 152 SAT scores with a normal distribution with a =500 and =100 In a random sample of n=25 students, what is the probability that the sample mean would be greater than 540? The primary purpose of the distribution of sample means is to fine the probability associated with any specific sample. Probabilities = proportions

Figure 7.3 A distribution of sample means Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

Z-scores for Sample Means Z-scores describe the position of any specific sample w/in the distribution The z-score for each distribution can be calculated using: z=X-  x We use the standard error instead of the standard deviation

General Concepts Standard error: samples will not provide perfectly accurate representations of the population Standard error provides a method for defining and and measuring sampling error. Individual sample means tend to overestimate or underestimate the population mean.

Figure 7.6 The structure of research study Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

Figure 7.8 Showing standard error in a graph Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning