Download Hypothesis Testing with a Distribution of Means

Chapter 6
Hypothesis Tests
with Means of
Samples
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
The Distribution of Means
• The distribution of means is
– A distribution of the means of a large number
of samples of the same size
– Each sample randomly selected from the same
population of individuals
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Characteristics of the
Distribution of Means
• Mean
– Same as the mean of the population of individuals
• Variance
– Variance of the population of individuals divided by the number of
individuals in each sample
– SD of the distribution of means is the square root of the variance of
the distribution of means
• Shape
– Approximately normal if either
• Each sample as at least 30 observations
• Underlying distribution of the population of individuals is normal
– In general, unimodal, symmetrical, less spread out than the
distribution of the population of individuals
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Review of Three Kinds of
Distributions
We’ve considered three kinds of distributions
A. The distribution of the population of
individuals
B. The distribution of a sample
C. The distribution of means of samples
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Hypothesis Testing with a
Distribution of Means
• Figuring Z score of a sample’s
mean on the distribution of
means
– Conceptually the same as figuring
a Z score for an individual on a
distribution of a population
– Subtract the mean of the
distribution of means from the
sample’s mean
– Then divide by the SD of the
distribution of means
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
First Example:
The first-year students of Shandong Normal University had
a test of computer literacy (The total score is 150). The mean
of the score was 100 (μ=100). The standard deviation was
15 (s=15). There were 70 students in educational technology
department. They got the mean of the score was 103.3.
Could we get the conclusion that the performance of
students in educational technology department was higher
than general first-year students of Shandong Normal
University?
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Estimation and
Confidence Intervals
• Point estimate
– Estimate of a specific value
of a population parameter
(e.g., the mean)
– The best estimate of the
mean of a population is the
mean of a sample taken
from that population
– SD of the distribution of
means (called the “standard
error”) is a measure of the
accuracy of the point
estimate
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Estimation and
Confidence Intervals
• Confidence interval
– Interval estimate with a
given likelihood of
including the population
mean
– Typical confidence intervals
• 95%
• 99%
– The larger the interval, the
greater one’s confidence
than the interval contains
the true mean
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Steps for Figuring 95% and
99% Confidence Intervals
• Estimate population
– Best estimate is sample mean
• Figure SD of distribution of means
• Find Z scores that go with confidence limits you
want
– For 95%, Z scores are +1.96 and -1.96
– For 99%, Z scores are +2.57 and - 2.57
• Convert Z scores to raw scores
– Multiply Z score by SD of distribution of means and
add to the population mean
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Hypothesis Testing
• Confidence intervals sometimes used instead of,
or alongside, ordinary hypothesis testing
– If confidence interval does not include the mean of the
null hypothesis distribution, result is statistically
significant
• Rare for researchers to know the characteristics of
the population distribution, but when they do, the
hypothesis test that is appropriate is a Z test.
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Second Example:
The standard deviation of computer literacy performance
at Shandong Normal University was 5. There was a class
which included 49 students at Shandong Normal
University. The average score of this class was 85.
Please inferred the confidence interval of computer
literacy performance about the students at Shandong
Normal University.
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall