1. Hypothesis Tests with Means of Samples
Chapter 6

2. 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

3. Distribution of Means Study a sample of many individuals
When N > 1, a special problem with Step 2
(Comparison distribution)
Problem: score in sample is the Mean of the group of scores.
Chapter 5: Distribution of population of individuals

4. Distribution of Means
Cannot Compare mean (N = 20) to distribution of a population of N =1
Interested in mean of a sample (N = 20)
Need comparison distribution that is a distribution of all possible means of samples of 20 scores.
A distribution of means (Sampling distribution of the mean):
Distribution of means of all possible samples (LOTS AND LOTS) of a given size drawn randomly from the population.

5. Building a Distribution of Means
Distribution of means: Randomly choosing samples of equal sizes from a population and take the means of those samples.
Those means 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

6. Population of
Happily Married
People
Choose
N = 4
= 26/4
Mean =6.25
Distribution
of Means
Plot Mean

7. Characteristics of a Distribution of Means
Characteristics of the comparison distribution that you need are:
Mean
SD2 & SD
the shape

8. Determine Characteristics of a Distribution of Means
The mean of the distribution of means is about the same as the mean of the original population of individuals (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 (true for all distributions of means).
Shape of the distribution of means is approximately normal (true for most distributions of means).

9. Mean of a Distribution of Means
The mean of a distribution of means (Population MM) 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
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.

10. SD2 and SD of the Distribution of Means
If each sample has N = 1, the distribution of means is the same as the population of individuals
When sample size is 2 or more, variance of the distribution of means is always smaller than the variance of the population.
This is because the probability of getting two or more extreme cases in the same direction is less than getting just one.
The more cases in each sample, the smaller the variation.

11. Distribution of Means
Larger N
Smaller N
Smallest N

12. Distribution of
the population
of Individuals
SDM
MeanM

13. clt

14. www. stat. ucla. edu/~dinov/courses_students. dir/05/Winter/STAT35
clt

15. Variance of a Distribution of Means
Population SD2M = Population SD2 N
If Pop SD2 = 9 and N= 3
Population SD2M = 9/3 = 3
Population SDM = 1.7
IF Pop SD = 3; N= 6 (NOTICE SD!!)
Population SD2M = 9/6 = 1.5
Population SDM = 1.14

16. 172
---- = 57.8 5

17. Standard Deviation of a Distribution of Means
Population SDM = √Population SD2M
Population SDM = standard deviation of the distribution of means
Population SDM (standard error of the mean).
How much the means in the distribution of means deviate from the mean of the population of individuals

21. Shape of a Distribution of Means
Shape of a distribution of means approximately normal if:
N ≥ 30 individuals or
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 & symmetrical.

22. 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

23. 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?

24. The Z Test
Z Test
Hypothesis-testing: With a single sample & population SD2 is known
Comparison distribution: Distribution of means.
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.

25. Figuring the Z Score of a Sample’s Mean on the Distribution of Means
Sample Mean = 25, Pop. MM = 15, and Pop. SDM = 5
the Z score of the sample’s mean would be?
Z = (M – Pop. MM)
Population SDM
Z = (25 – 15) = 2 5

26. Example
A person says that she can identify people of above average intelligence with her eyes closed (N =20).
STEP1. Reframe question into a research and null hypothesis about populations.
Population 1: People chosen by woman with her eyes closed (THIS IS THE SAMPLE)
Population 2: People in general (Comparison Distribution, KNOWN)

27. Research Hypothesis (HA): Those chosen are more intelligent
Population 1 > Population 2
Population 1 has a higher mean intelligence than population 2
Null Hypothesis (H0): Those chosen are not more intelligent
Population 1 = Population 2
Population 1 does not have higher mean intelligence than population 2.

28. STEP 2. Determine Characteristics of the comparison distribution
Known normal distribution with Population M = 100; Pop. SD = 16
Distribution of means, Normal
Pop. MM = 100, Pop. SD2 = 256 Why 256?
Pop. SD2M = Pop. SD2/N= 256/20 = 12.8; Pop. SDM= 3.58
STEP 3. Determine the Cutoff Sample scores on the comparison distribution at which the null hypothesis should be rejected (p < .0001)
.01% probability (top .01%), Z needed is 3.719
50% = % >3.719

30. Person picks out 20 individuals; IQ score = 140
STEP 4. Determine the score of your sample on the comparison distribution
Person picks out 20 individuals; IQ score = 140
Z= (M – Pop. MM)/Pop. SDM
(140 – 100)/3.58 = 40/3.58 = 11.17
5. Compare the scores obtained in Step 3 and 4 to decide whether to reject the null hypothesis
Score on 4 (Z = 11.17) vs. score on 3 (Z = 3.719)
Conclusion: Reject H0: Research hypothesis is supported

38. 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.
Copyright © 2011 by Pearson Education, Inc. All rights reserved

39. 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.

40. 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.