Chapter 7
Hypothesis Tests With
Means of Samples
The Distribution of Means
 Comparison distributions considered so
far were distributions of individual
scores
 Mean of a group of scores
– Comparison distribution is distribution of
means
The Distribution of Means
 Distribution of means
– Distribution of the means of each of a very
large number of samples of the same size
(with each sample randomly taken from the
same population of individuals)
The Distribution of Means
 Characteristics
– Its mean is the same as
the mean of the population
of individuals
– Its variance is the variance
of the population divided by
the number of individuals in
each of the samples
M  

 
N
2
2
M
The Distribution of Means
 Characteristics
– Its standard deviation is the square root of
its variance
M  
2
M
2



N
N
– Shape: it is approximately normal if either
• Each sample is of 30 or more individuals or
• The distribution of the population of individuals
is normal
Review of the Different
Kinds of Distributions
 Distribution of a population of individuals
 Distribution of a particular sample
 Distribution of means
Comparison of Three Types
of Distributions
Hypothesis Testing With a
Distribution of Means
 It is the comparison distribution when a
sample has more than one individual
 Find a Z score of your sample’s mean
on a distribution of means
(M   M )
Z
M
Estimation, Standard Errors,
and Confidence Intervals
 Estimating the mean: point estimates
 Accuracy of a point estimate
 Interval estimates
– Confidence limits
– 95% confidence interval
– 99% confidence interval
Estimation, Standard Errors,
and Confidence Intervals
 Steps for figuring confidence limits
1. Figure the standard error
2
M 
N
2. Figure the raw scores for 1.96 standard
errors (95% confidence interval) or 2.57
standard errors (99% confidence interval)
above and below the sample mean
Estimation, Standard Errors,
and Confidence Intervals
 Subtle logic of hypothesis testing
 Confidence intervals and hypothesis
testing
Controversies and Limitations
 Confidence intervals or significance
tests?
– Confidence intervals
• Give additional information
• Focus attention on estimation
• Less likely to be misused by researchers
– Significance tests
• Necessary for some advanced statistical
procedures
Reporting in Research
Articles
 Z test
 Standard error, SE, SEM
– Standard error bars