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University of Guelph

Psychology

PSYC 2040

David Stanley

Winter

Description

Introduction
Statistic - a number that describes the sample
Parameter - a number that describes the population
Unbiased Estimate - the mean of all possible values of that
statistic is equal to the parameter
Biased Estimate - the mean of all possible values of that
statistic is not equal to the parameter
Population Standard Deviation - the amount that a score differs
from the population mean
Standard Error of the Mean - the amount that sample means differ
from the population mean due to the effects of random sampling,
larger sample sizes result in smaller standard errors
Degrees of Freedom - the number of values in a data set that are
allowed to vary, the first value is the between groups degrees
of freedom while the second is the within groups degrees of
freedom
Central Limit Theorem - regardless of the shape of the
population distribution, the distribution of the sample means
will always be normal as long as we have large sample sizes and
the standard error will equal the population standard deviation
divided by the square root of the sample size
Statistics is always about ratios, more specifically:
Single Sample T-tests
We use this when we are comparing one mean to another number.
Null Hypothesis: μ = 15; any difference between the sample mean
and the hypothesized population mean is due to random sampling.
Alternative Hypothesis: μ ≠ 15; any difference between the sample
mean and the hypothesized population mean is too large to be due
to random sampling alone therefore they must be different.
We make two assumptions when conducting a simple t-test: (a)
that there was random sampling from the population and (b) that
the population values are normally distributed. Our p-value tells us the probability of you finding the t-value
you did when the population mean is equal to the hypothesized
population mean.
When p = .00, there is a 0% chance of obtaining that t-
value due to random sampling when a population mean is
equal to the hypothesized one. Therefore we reject the null
hypothesis because there is less than a 5% chance of
getting that t-value.
We have to be careful because sometimes our alternative
hypothesis might be that it is greater than or less than a
value (one tailed test) while SPSS gives p-values for 2-
tailed tests. This means we would have to divide the p-
value by two to get the p-value for a one sided test.
The t-value tells you that the difference between the
hypothesized population mean and the real population mean is ___
bigger than you would expect if there was only random sampling
effects involved. Typically, the larger the t-value, the smaller
the p-value and this hold for the other test statistics as well.
The degrees of freedom are calculated by: number of participants
- number of groups.
In conclusion, we do a single sample t-test when there is only one population and
it is being compared to a hypothesized mean. There is 1 independent variable
with 2 levels, which are categorical and 1 dependent variable that is continuous.
Independent Samples T-test
We do this when we have two populations and we are comparing
their means. The difference between two sample means is used to
estimate the difference between the two population means.
Null Hypothesis: μ 1= μ 2 assume that mean on dependent variable
for population 1 is equal to mean on dependent variable for
population 2. Any difference seen between the two sample means
is due to random sampling alone.
Alternative Hypothesis: μ 1 ≠ μ2; the difference we see with the
sample means is too large to be due to just random sampling.
Conclude that mean on dependent variable for population 1 is
different for mean on dependent variable for population 2.
We make three assumptions when conducting an independent samples
t-test: (a) that there was random sampling of the two
populations, (b) that the population values are normally distributed and (c) that the variances of the two populations
are the same.
Since we are dealing with two populations we have to conduct a
Levene’s test for homogeneity of variance. This is because when
we conduct an F- or t-test we are looking to see if the between
group variance is larger than the within groups variance and to
do this ratio we can only include one within groups variance on
the bottom. Therefore it becomes:
We need to conduct a Levene‟s test to make sure it is okay to
use either or as the denominator. Our null hypothesis is that
the two variances are equal; σ 1= σ 2 while our alternative
hypothesis is that the two variances are not equal; σ 1 ≠ σ 2 An
independent samples t-test is the only time when we can continue
if the Levene‟s test is significant. In any other test if the
Levene‟s test is significant we cannot go any further in our
analysis.
Depending on if the Levene‟s test is significant or not we use a
different line. If significant we use the “equal variances not
assumed” line, if non-significant we use the “equal variances
assumed” line. The degrees of freedom for Levene‟s test are:
(number of groups -1, number of participants - number of
groups).
The degrees of freedom are: number of participants - number of
groups as long as Levene‟s test is non-significant.
Effect size tells us how big the difference is between the
populations. We use Cohen‟s d. Our p-value only tells us that
there is a difference and only allows you to reject the null
hypothesis for d-values greater than zero (there is a
difference). P-values do not tell us about the size of the
effect, that is what reporting an effect size does. If this is
not give on output do not report it.
Small = .20
Medium = .50
Large = .80
In conclusion, we do an independent sample t-test when there are two
populations that we are comparing. There is 1 independent variable with 2
levels, which are categorical and 1 dependent variable that is continuous.
Independent samples t-test is the only time we can continue when Levene’s test
is significant. Repeated Measures T-test
We use a repeated measures t-test when we are measuring the
dependent variable on the same person, twice.
Null Hypothesis: μ Difference 0; the difference between the two
conditions is zero. Any difference we see is due to the effects
of random sampling. Assume that the sample is taken from a
population with a mean difference of 0.
Alternative Hypothesis: μ Difference0; the difference between the
two conditions is too large to be due to the effects of random
sampling alone.
We make two assumptions when conducting a repeated measures t-
test: (a) that there was random sampling from the population and
(b) that the population difference values are normally
distributed.
There is no Levene’s test for a repeated measures t-test because
there is only 1 population.
Degrees of freedom are: number of participants - number of
groups. In this case, it will always be minus one because there
is one group being measured twice.
In conclusion, we do a repeated measures t-test when there is one population
that is being measured on the dependent variable twice. There is 1 independent
variable with 2 levels, which are categorical and 1 dependent variable that is
continuous.
One-Way ANOVA
We use a one-way ANOVA test when we are measuring the effects of
an independent variable with 3+ levels on a dependent variable.
Why don‟t we use three separate t-tests? Because this increases
the chance of Type 1 error (probability of rejecting the null
hypothesis when it is actually true) to .15. A one-way ANOVA
keeps this value at .05.
Null Hypothesis: μ Population 1μPopulation 2 μPopulation 3 mean score
is the same for all populations and any difference we see is due
to random sampling. Alternative Hypothesis: Not all population means are the same;
the differences among sample means are too large to be due to
just random sampling alone.
In ANOVAs we calculate an F test statistic. We start by assuming
all populations have the same variance (σ) even though we expect
them to be different. Even if they were to be the same, random
sampling would result in sample means that are somewhat
different. With the F test statistic we are looking to compare
the observed variability in sample means and the variability we
would expect due to random sampling.
If the populations have identical means, the observed
variability in sample means should be roughly equal to the
expected variability. Therefore,
If the populations have different means, the observed
variability in sample means should be larger than the expected
variability. Therefore,
If the p-value is less than .05 there is less than a 5% chance
that we could obtain an F-value that we did if the population
means were actually equal. Therefore we reject the null and say
that they aren‟t. The degrees of freedom for Levene‟s test are:
(number of groups - 1, number of participants - number of
groups).
Our null hypothesis for Levene‟s test is that σ Population 1
σPopulation 2σPopulation 3and our alternative hypothesis is that
not all standard deviations are the same. We cannot continue our
analysis if Levene‟s test is significant. When non significant,
we can pool the sample standard deviations together to get a
better estimate of the population standard deviation. This
number is also called the Mean Square Error and can be read off
the output. If Levene‟s test was significant we would have no
choice but to do a series of t-tests.
There are four assumptions we make when we conduct a one-way
ANOVA test: (a) observations represent random samples of
populations, (b) observations are from normally distributed
populations, (c) population variances are equal and (d) the
numerator and denominator of the F-ratio are independent. We assume a, b, and d are true and use Levene‟s test to see if c is
true. If we fail our assumptions: (a) conclusions don‟t
generalize, (b) F distribution is not terribly affected b small
departures, (c) only assumption we test, can be a problem if
variances are extremely different and (d) don‟t have subject‟s
scores be dependent on one another.
If our F-value is significant we say that the observed variance
between the sample means was ___ times larger than we would have
expected if due to random sampling if the means were the same.
Therefore we conclude that the population means are not the
same. The degrees of freedom for an F-test are: (number of
groups - 1, number of participants - number of groups). These
are the same numbers as for the Levene‟s test.
The effect size used for F-tests is partial η . With this
statistic:
Small = .01
Medium = .06
Large = .14
The F-test only tells us that there is a difference between the
population means but it does not tell us where. We then use
posthoc analyses to determine what pairs are different and they
are used when you don‟t have a specific prediction about
particular groups being higher / lower than others. To control
for Type 1 error like before we use one of 2 posthoc tests:
1. Bonferroni Correction for t-tests
a. We divide .05 by the number of populations and use the
resulting value to determine significance. If there
were 3 populations, we would divide .05 by 3 to get
.017 and conduct 3 t-tests (one for each possible
combination) and deem the comparison significant if
the p-value was less than .017.
b. We would still report that p < .05.
2. Tukey Honestly Significant Difference (HSD)
a. We use a q-test to ensure Type 1 error does not exceed
.05. This is only used if: (a) the main effect is
significant [the one-way ANOVA is significant] and (b)
if the independent variable has 3+ levels. You then
compare each possible combination of levels and see if
you get a significant result. The significance values
are found by typing the degrees of freedom (number of
means, dferror into a website.
i. If q > Q 0.05then it is significant at the .05
level. ii. If q > Q 0.01then it is significant at the .01
level.
b. Consequently this shows us where the differences lie.
Lastly, we make a bar graph to display our results. On the x-
axis we have the independent variable with each level being it‟s
own bar, and on the y-axis the dependent variable. The mean
values of the dependent variable you plot can be read off of the
output. Remember that figure descriptions are at the bottom of
the graph and „figure‟ is in italics.
Figure 1. Mean rating values representing the average attractiveness rating for each of the
perfumes.
CRF ANOVA
In conclusion, we do a one-way ANOVA when there is 1 independent variable
with 3+ levels, which are categorical and 1 dependent variable that is continuous.
We conduct a Levene’s test but it must be non-significant to continue. A bar
graph is also made.
We conduct a completely randomized factorial (CRF) ANOVA when
there are 2 or more independent variables with multiple levels
that are categorical and 1 dependent variable that is
continuous.
Main effects look at marginal means to see if they are the same
and it is the single effect of a factor on the dependent
variable while ignoring the other factor and it is calculated
for each independent variable. We have a null and alternative hypothesis for each factor. Main effect degrees of freedom are:
(number of levels in factor -1, number of participants - number
of cells)
Factor 1
o Null Hypothesis: μ A = μB = μC; therefore any variability
in our sample means is due to random sampling.
Population means are equal and any variability is due
to random sampling, F=1.
o Alternative Hypothesis: not all population means are
the same; therefore any variability in our sample
means is too big to be due to chance alone. Population
means and the variability mean means that F >1.
Factor 2
o Null Hypothesis: μ 1 = μ2 = μ3; therefore any variability
in our sample means is due to random sampling.
Population means are equal and any variability is due
to random sampling, F=1.
o Alternative Hypothesis: not all population means are
the same; therefore any variability in our sample
means is too big to be due to chance alone. Population
means and the variability mean means that F >1.
Etc.
An interaction looks at the combined effect of the factors on
the dependent variable and a significant interaction says that
the means do not act consistently across factor 1 at every level
of factor 2 or vice versa. It means that the effect a factor has
on the dependent variable depends on the level of the second
factor.
Null Hypothesis: No Interaction; the effect of one
independent variable on the dependent variable does not
differ at the various levels of the other independent
variable.
Alternative Hypothesis: Interaction; the effect of one
factor varies depending on the level of the other

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