1. Chapter 7 Probability and Samples
PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J. Gravetter and Larry B. Wallnau

2. 7.1 Samples and Population
Chapter 5 taught us that the location of a score (an X value) can be represented with a z-score.
However, researchers typically study large samples rather than single scores because samples provide better estimates of the population.
We can transform our sample mean to a z-score, just like we did with an X value.
But remember, with sample means we are dealing with samples so we always have to be aware of sampling error (with raw scores, we don’t)

3. Sampling Error “Error” here does not indicate a mistake was made.
Sampling error is the natural discrepancy, or the amount of error, between a sample statistic and its corresponding population parameter.
Samples are variable; no two samples are identical…they vary from one to another.

4. 7.2 Distribution of Sample Means
Samples differ from each other, therefore the sample means differ from each other.
The distribution of sample means is the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population.
In reality, this is not a real, measured distribution---it represents the distribution IF we could obtain all samples from a population. It is imaginary.

5. Sampling distribution
Distributions in earlier chapters were distributions of raw scores (X values).
Distribution of sample means is called a sampling distribution:
A distribution of statistics obtained by selecting all the possible samples of a specific size (n) from a population
So, the distribution of sample means (a statistics) is an example of a sampling distribution—it is called the sampling distribution of M

6. Figure 7.1 Frequency distribution histogram for population of scores

7. Figure 7.2 Distribution of sample means

8. Characteristics of distributions of sample means
The sample means will pile up around the population mean.
The distribution of sample means is approximately normal in shape.
See where we are going with this??

9. Central Limit Theorem
Applies to any population (mean μ and standard deviation σ)
Distribution of sample means approaches a normal distribution as n approaches infinity
Distribution of sample means for sample of size n will have a mean of μ
Distribution of sample means for sample of size n will have a standard deviation of

10. Shape of distribution of sample means
The distribution of sample means is almost perfectly normal in either of two conditions
The population from which the samples are selected is a normal distribution or
The number of scores (n) in each sample is relatively large, around 30 or more.

11. Expected value of M
Mean of distribution of sample means is always equal to the population mean, μ.
Mean of the distribution of sample means is called the expected value of M.
M is an unbiased statistic.

12. Standard error of M Variability is measured by standard deviation.
The variability of sample means is measured by the standard deviation of the all of the sample means
This is called the standard error (σM ) and computed as σM =

13. Magnitude (size) of Standard Error
Law of large numbers: the larger the sample size, the more probable it is that the sample mean will be close to the population mean.
Population variance: The greater the variance in the population, the less probable it is that the sample mean will be close to the population mean.

14. Three Different Distributions

15. Learning Check
A normal population has μ = 60 with σ = 5. The distribution of sample means for samples of size n = 4 selected from this population would have an expected value of _____. A 5 B 60 C 30 D 15

16. Learning Check
A normal population has μ = 60 with σ = 5. The distribution of sample means for samples of size n = 4 selected from this population would have an expected value of _____. A 5 B 60 C 30 D 15

17. Learning Check
Decide if each of the following statements is True or False.
T/F
The distribution of sample means is always normal shaped
As sample size increases, the value of the standard error decreases.

18. Answer It is normal shaped if the population is normal or n ≥ 30
False
It is normal shaped if the population is normal or n ≥ 30
True
The sample size is in the denominator of the equation

19. 7.3 Probability and the distribution of sample means
The primary use of the distribution of sample means is to find the probability associated with any particular sample.
Proportions of the normal curve are used to represent probabilities.
A z-score for sample means is computed.
Just like we found probabilities associated with a particular score.

20. Figure 7.5 Distribution of sample means for n = 25

21. Z-score for sample means
Z = M – μ σm
M = μ + Z(σm)

22. Example 7.3 Proportions of distribution of sample means.

23. Example of typical distribution of sample means

24. 7.4 More about standard error
There will be discrepancy between a sample mean and the true population mean
This is sampling error
The amount of sampling error varies
The variability of sampling error is measured by the standard error—this is the average distance of all the sample means from the population mean.
Standard error is the same thing as standard deviation---it just applies to the sampling distribution instead of the raw score distribution.

25. Example 7.4 Distribution of sample means for samples of sizes 1, 4, and 100

26. Mean (± SE) in bar chart

27. Mean (± SE) in line graph

28. 7.5 Looking ahead to inferential statistics
Inferential statistics use sample data to draw general conclusions about populations
Sample is not perfectly accurate reflection of its population
Differences between sample and population introduce uncertainty into inferential processes
Statistical techniques use probabilities to draw inferences from sample data

29. Structure of research study Example 7.5

30. Figure 7.12 Distribution of sample means in Example 7.5

31. Learning Check
A random sample of n = 16 scores is obtained from a population with µ = 50 and σ = 16. If the sample mean is M = 58, what is the z-score corresponding to the sample mean? A
z = 1.00 B
z = 2.00 C
z = 4.00 D
Cannot determine with more information
S = 16/4 = 4 so z = 58-50/4 = 2

32. Learning Check
A random sample of n = 16 scores is obtained from a population with µ = 50 and σ = 16. If the sample mean is M = 58, what is the z-score corresponding to the sample mean? A
z = 1.00 B
z = 2.00 C
z = 4.00 D
Cannot determine with more information

33. Learning Check
Decide if each of the following statements is True or False.
T/F
A sample mean with z = 3.00 is a fairly typical, representative sample
The mean of the sample is always equal to the population mean

34. Answer Individual samples will vary from the population mean
False
A z-score of 3.00 is an extreme, or unlikely, z-score
Individual samples will vary from the population mean

35. Any Questions?
Concepts?
Equations?