Probability and Samples: The Distribution of Sample Means Chapter 7.

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

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

• 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

Figure 7.1

Frequency distribution for a population of four scores: 2, 4, 6, 8

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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

Table 7.1

The possible samples of n = 2 scores from the population in Figure 7.1

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Figure 7.2Figure 7.2

The distribution of sample means for n = 2

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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.

nnnn

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)

X

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

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.

Learning Check pg 151

1) A population of scores is normal with =80 and =20

a) 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)

b) How would the distribution of sample means be changed if the sample size were n=100 instead of n=16.

• 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?

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?

Figure 7.3

A distribution of sample means

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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

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.

Figure 7.6

The structure of research study

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Figure 7.8

Showing standard error in a graph

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