Chapter 5 – Hypothesis Tests with Means of Samples 1
The Distribution of Means
With a larger sample, there is a problem with determining the characteristics of the comparison distribution (CD). In the
previous chapter (samples of 1) the CD has been a distribution of individual scores. This chapter deals with larger samples.
The score you care about most is the mean of a group of scores.
Suppose you were to compare the mean of this sample of 64 individuals’ scores to a distribution of a population of
individual scores. This is like comparing apples with oranges: wrong.
When you are interested in the mean of a sample of 64 scores, you need a comparison distribution that is a
distribution of means of samples of 64 scores.
o Distribution of means – distribution of means of samples of a given size from a population; comparison
distribution; the scores in a distribution of means are means, not scores of single individuals
Determining the Characteristics of a Distribution of Means
Step 2 of hypothesis testing involves determining the characteristics of the comparison distribution. The three key
characteristics of the comparison distribution that you need to determine are its mean, spread (using δ and δ )
1. Its mean
2. Its spread (measure using variance and standard deviation)
3. Its shape
Three Rules about the Distribution of Means:
Rules Definition Equation Notes
The mean of a sample will
Mean of a distribution of µ M µ sometimes be higher and
1 – MEAN means – the mean of a sometimes lower than the
mean of the whole population
The mean of a distribution of distribution of means of µM=mean of a distribution of individuals. Since the
means is the same as the samples of a given size from of means selection process is random
mean of the population of a population; comes out to
individuals (µ) be the same as the mean of and there is a large # of
the population of individuals µ = mean of the samples, the high and low
population of individuals means eventually balance each
δ2 = variance of a distribution
2A – SPREAD Variance of a distribution of M
The variance of a distribution means – variance of the o2 means
of means is the variance of δ = variance of the population
the population of individuals population divided by the of individuals
divided by the # of individualsnumber of scores in each N = # of individuals in each
in each sample. sample
2B – SPREAD Aka standard error of the mean
Standard deviation of a (SEM) or standard error (EM)
The standard deviation of a distribution of means – Tells you how much the various
distribution of means is the square root of the variance of means in the distribution of
square root of the variance of
the distribution of means a distribution of means means tend to deviate from
the mean of the population
3 – SHAPE
The shape of a distribution of means is approximately normal if either (a) each sample is of 30 or more individuals or (b) the
distribution of the population of individuals is normal.
The rules are based on the central limit theorem that provides the key characteristics of a distribution of means for
a population with a distribution of any shape. The characteristics are: central tendency, variability, and shape. Chapter 5 – Hypothesis Tests with Means of Samples 2
THREE T YPES OFDISTRIBUTIONS:
Population’s Particular Sample’s
Distribution Distribution Distribution of Means
Scores of all individuals Scores of the individuals in Means of samples
Content randomly taken from the
the population a single sample population
Approximately normal if
Any shape but often samples have 30
Shape normal Any shape individuals in each or if
population is normal
M = (X)/N m=
Variance 2 SD = [(X – M) ]/N 2M = /N
Standard deviation SD = M=
Hypothesis Testing with a Distribution of Means: The Z Test
Z test – hypothesis-testing procedure in which there is a single sample and the population variance is known
When a researcher studies a sample of more than one person, the distribution of means IS the comparison
distribution. It is the distribution to which you compare your sample’s means. This comparison tells you how likely
it is that you could have selected a sample with a mean that is this extr0me if the H is true.
Figuring the Z Score of a Sample’s Mean on the Distribution of Means
You are finding a Z score of your sample’s mean on a distribution of means. In previous chapters we had to find the
Z score of a single individual on a distribution of single individuals. Treat the sample mean like a single score.
Ordinary formula for changing a raw score to a Z score: Z = (X – M)/SD
Formula for the present situation
The Z score for the sample’s mean on the distribution of means is the sample’s mean minus the mean of
the distribution of means, divided by the standard deviation of the distribution of means.
Your sample’s mean is 18 and the distribution of means has a mean of 10 and a standard deviation of 4.
ONE – Restate the question as a research hypothesis and a null hypothesis about the populations.
The two populations are:
1 Students who are told that the person has positive personality qualities.
2 Students in general who are told nothing about the person’s personality qualities. Chapter 5 – Hypothesis Tests with Means of Samples 3
The research hypothesis would be that the
population of students who are told that the person The null hypothesis is that Population 1’s scores will
has positive personality qualities will on average give not on average be higher than Population 2’1
higher attractiveness scores for that person than the 2
population of students who are told nothing about
the person’s personality qualiti1:> 2
**These are directional hypotheses, meaning the researcher wants to know if being told that the person has
positive personality qualities will increase attractiveness scores. If they received a result in the opposite direction,
it would not be relevant to the theory they are testing.
TWO – Determine the characteristics of the comparison distribution
The result will be a mean of a sample of 64 individuals (students in this example), meaning the CD has to be the
distribution of means of samples of 64 individuals each (as in multiple groups of 64 students).
The mean will be the same as the population mean, 200(M= 200).
The variance will be the population variance divided by the number of individuals in the sample.
o Population variance 2) is 2304 and the sample size is 64.