1. Stony Brook University
Statistics for the Behavioral and Social Sciences: A Brief Course Fifth Edition Arthur Aron, Elaine N. Aron, Elliot Coups
Prepared by:
Genna Hymowitz
Stony Brook University
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2. Hypothesis Tests with Means of Samples
Chapter 6
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3. Chapter Outline The Distribution of Means
Hypothesis Testing with a Distribution of Means: The Z Test
Hypothesis Tests about Means of Samples (Z Tests) in Research Articles
Advanced Topic: Estimation and Confidence Intervals
Advanced Topic: Confidence Intervals in Research Articles
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4. The Distribution of Means
The score you care about when you have a sample of more than one person is the mean of the group of scores.
The comparison distribution in which you are interested is a distribution of means of a sample of a given size from a particular population.
The scores in a distribution of means are means, not scores of individuals.
A distribution of means is a distribution of the means of each of lots and lots of samples of the same size.
Each sample is randomly taken from the same population of individuals.
The distribution of means is the correct comparison distribution when there is more than one person in a sample.
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5. Building a Distribution of Means
Think of a distribution of means as if you kept randomly choosing samples of equal sizes from a population and took the means of those samples.
Those means are what make up a distribution of means.
The characteristics of a distribution of means can be calculated from:
characteristics of the population of individuals
number of scores in each sample
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6. Determining the Characteristics of a Distribution of Means
Characteristics of the comparison distribution that you need are:
the mean
the variance and standard deviation
the shape
The mean of the distribution of means is about the same as the mean of the original population of individuals.
This is true for all distributions of means.
The spread of the distribution of means is less than the spread of the distribution of the population of individuals.
The shape of the distribution of means is approximately normal.
This is true for most distributions of means.
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7. Mean of a Distribution of Means
The mean of a distribution of means of samples of a given size from a particular population
It is the same as the mean of the population of individuals.
Population MM = Population M
Population MM is the mean of the distribution of means.
Because the selection process is random and because we are taking a very large number of samples, eventually the high means and the low means perfectly balance each other out.
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8. Variance of a Distribution of Means
The variance of a distribution of means is the variance of the population of individuals divided by the number of individuals in each sample.
Population SD2M = Population SD2 N
Population SD2M = the variance of the distribution of means
Population SD2 = the variance of the population of individuals
N = number of individuals in each sample .
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9. Standard Deviation of a Distribution of Means
The standard deviation of a distribution of means is the square root of the variance of the distribution of means comparison distribution.
Population SDM = √Population SD2M
Population SDM = standard deviation of the distribution of means
Population SDM is also known as the standard error of the mean.
tells you how much the means in the distribution of means deviate from the mean of the population
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10. The Shape of a Distribution of Means
The shape of a distribution of means is approximately normal if either:
each sample is of 30 or more individuals or
the distribution of the population of individuals is normal
Regardless of the shape of the distribution of the population of individuals, the distribution of means tends to be unimodal and symmetrical.
Middle scores for means are more likely and extreme means are less likely.
A distribution of means tends to be symmetrical because lack of symmetry is caused by extremes.
Since there are fewer extremes in a distribution of means, there is less asymmetry.
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11. Review of the Three Kinds of Distributions
Population’s Distribution
made up of scores of all individuals in the population
could be any shape, but is often normal
Population M represents the mean.
Population SD2 represents the variance.
Population SD represents the standard deviation.
Particular Sample’s Distribution
made up of scores of the individuals in a single sample
could be any shape
M = (∑X) / N calculated from scores of those in the sample
SD2 = [∑(X – M)2] / N
SD = √SD2
Distribution of Means
means of samples randomly taken from the population
approximately normal if each sample has at least 30 individuals or if population is normal
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12. How Are You Doing?
Why is a distribution of means used when evaluating a sample of more than one individual?
What are the guidelines for finding the characteristics of a distribution of means?
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13. Hypothesis Testing with a Distribution of Means: The Z Test
Hypothesis-testing procedure in which there is a single sample and the population variance is known
The comparison distribution for the Z test is a distribution of means.
The distribution of means is the distribution to which you compare your sample’s mean to see how likely it is that you could have selected a sample with a mean that extreme if the null hypothesis were true.
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14. Figuring the Z Score of a Sample’s Mean on the Distribution of Means
If you had a sample with a mean of 25, a distribution of means with a mean of 15, and a standard deviation of 5, the Z score of the sample’s mean would be 2.
Z = (M - Population MM)
Population SDM
Z = (25 – 15) = 2 5
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15. Steps for Hypothesis Testing
The steps for hypothesis testing are the same for a sample of more than 1 as they are for a sample of 1.
Step 1: Restate the question as a research hypothesis and a null hypothesis about the population.
Step 2: Determine the characteristics of the comparison distribution.
Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
Step 4: Determine your sample’s score on the comparison distribution.
Step 5: Decide whether to reject the null hypothesis.
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16. Example of Steps for Hypothesis Testing: Step 1
Students give a standard school achievement test to 64 randomly selected fifth-grade school children.
The test is given in the usual way, except that they add to the instructions a statement that children are to answer each question with the first response that comes to mind.
When given in the usual way the mean is 200, the standard deviation is 48, and the distribution is approximately normal.
Step 1: Restate the question as a research hypothesis and a null hypothesis about the population.
Population 1: fifth graders who get the special instructions
Population 2: fifth graders in general (who do not get the special instructions)
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17. Example of Steps for Hypothesis Testing: Step 2
Step 2: Determine the characteristics of the comparison distribution.
The comparison distribution is a distribution of means of samples of 64 individuals each.
The mean is 200 (the same as the population mean).
The variance will be the population variance.
Population SD2 = 2.304, sample size = 64
Population SD2M = / 64 = 36
Population SDM = √36 = 6
The shape of the distribution will be approximately normal because the sample size is larger than 30.
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18. Example of Steps for Hypothesis Testing: Step 3
Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
With a 5% significance level and a directional hypothesis, the null hypothesis will be rejected if the result is in the top 5% of the comparison distribution.
You can use the normal curve table to find the cutoff score.
The top 5% starts at a Z score of
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19. Example of Steps for Hypothesis Testing: Step 4
Step 4: Determine your sample’s score on the comparison distribution.
If in this example the mean of the sample was 220:
Z = (M – Population MM) / Population SDM
Z = (220 – 200) / 6 = 20 / 6 = 3.33
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20. Example of Steps for Hypothesis Testing: Step 5
Step 5: Decide whether to reject the null hypothesis.
The sample’s Z score is 3.33, which is higher than the cutoff score of 1.64.
The researchers can reject the null hypothesis.
The results of the study are statistically significant at the p < .05 level.
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22. Hypothesis Tests about Means of Samples in Research Articles
Z tests are not often seen in research articles because it is rare to know a population’s mean and standard deviation.
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23. Advanced Topic: Estimation and Confidence Intervals
Estimating the population mean based on the scores in a sample is an important approach in experimental and survey research.
When the population mean is unknown, the best estimate of the population mean is the sample mean.
The accuracy of the population mean estimate is the standard deviation of the distribution of means (standard error).
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24. Range of Possible Means Likely to Include the Population Mean
Confidence Interval
used to get a sense of the accuracy of an estimated population mean
It is the range of population means from which it is not highly unlikely that you could have obtained your sample mean.
95% confidence interval
confidence interval for which there is approximately a 95% change that the population mean falls in this interval
Z scores from to on the distribution of means
99% confidence interval
confidence interval for which there is approximately a 99% chance that the population mean falls in this interval
Z scores from to +2.58
confidence limit
upper and lower value of a confidence interval
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25. Figuring the 95% and 99% Confidence Intervals
Estimate the population mean and figure the standard deviation of the distribution of means.
The best estimate of the population mean is the sample mean.
Find the variance of the distribution of means.
Population S2M = Population SD2 / N
Take the square root of the variance of the distribution of means to find the standard deviation of the distribution of means.
Population SDM = √Population SD2M
Find the Z scores that go with the confidence interval you want.
95% CI Z scores are and -1.96
99% CI Z scores are and -2.58
To find the confidence interval, change these Z scores to raw scores.
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26. Example of Figuring the 99% Confidence Interval
If we used the earlier example of 64 fifth graders who received special instructions for a test:
The population mean is 200 and the standard deviation is 48.
The sample mean is 220.
Estimate the population mean and figure the standard deviation of the distribution of means.
The best estimate of the population mean is the sample mean of 220.
Find the variance of the distribution of means.
Population S2M = Population SD2 / N = 482 / 64= 36
Take the square root of the variance of the distribution of means to find the standard deviation of the distribution of means.
Population SDM = √Population SD2M = √36 = 6
Find the Z scores that go with the confidence interval you want .
99% CI Z scores are and -2.58
To find the confidence interval ,change these Z scores to raw scores.
lower limit = (-2.58)(6) = =
upper limit = (+2.58)(6) = =
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27. Confidence Intervals In Research Articles
Confidence intervals are becoming more common in research articles in some fields.
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28. Key Points
A distribution of means is the comparison distribution when studying a sample of more than one individual.
The distribution of means has the same mean as the corresponding population of individuals. The variance of the distribution of means is the variance of the population of individuals divided by the number of individuals in each sample. Its standard deviation is the square root of its variance. The shape of a distribution of means is close to normal if the number of participants in the samples is at least 30 or if the population of individuals follows a normal curve.
Z tests are hypothesis tests with a single sample of more than one individual and a known population. The comparison distribution for Z tests is a distribution of means.
Z tests are rarely found in research articles.
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29. Key Points for Advanced Topic: Confidence Intervals
The sample mean is the best estimate for the population mean when the population mean is unknown. The accuracy of the estimate is the standard deviation of the distribution of means.
Confidence intervals are ranges of possible means that are likely to include the population mean.
For a 95% confidence interval, the Z score range is to
For a 99% confidence interval, the range is from standard deviations to standard deviations.
Confidence intervals are sometimes reported in research articles.
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