Download The Nature and Logic of Probability

Chapter Nine
Probability, the Normal
Curve, and Sampling
PowerPoint Presentation created by
Dr. Susan R. Burns
Morningside College
Smith/Davis (c) 2005 Prentice Hall
The Nature and Logic of Probability
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Probability – is used to help determine the likelihood that a
given outcome from out experiment was probable or not.
Anytime we use a sample and not a population, we have to use
statistics and probability to draw conclusions.
– Subjective Probability – is used in estimations of probability,
not using numbers.
– Odds – when you hear about the odds of something taking
place using numbers; that is a type of probability.
– Percentages – are a common way to use numbers to
denote probability.
– Proportions – also use numbers to indicate probability.
Smith/Davis (c) 2005 Prentice Hall
The Nature and Logic of Probability
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The reason probability is so important for
researchers is that inferential statistics tests allow us
to determine the probability that our results came
about by chance.
The basic premise behind inferential statistics is that
we assume there is no difference between the
groups in our experiment (a.k.a., the null
hypothesis).
Smith/Davis (c) 2005 Prentice Hall
The Nature and Logic of Probability
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With two groups the null hypothesis looks like this:
M1 = M2
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In other words, the mean of group 1 is equal to the mean of
group 2.
The second hypothesis we test in inferential
statistics is the alternative (or experimental)
hypothesis. Written as:
M1  M2
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The alternative hypothesis is typically the one the
experimenter hopes to support in the experiment. To avoid
bias, we start off assuming the null hypothesis and let the
data and statistical test demonstrate otherwise.
Smith/Davis (c) 2005 Prentice Hall
The Nature and Logic of Probability
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If the differences between the groups is small or zero, our
inferential statistic would also be small.
Inferential statistics with small values occur frequently by
chance.
If it occurs by chance, then we say that the difference between
our groups is not significant and conclude that our IV did not
affect the DV, and thus accept the null hypothesis.
If the difference between the groups is large, our inferential
statistic would be large.
Inferential statistics with large values occur rarely by chance,
and thus, we would say that the difference between our group
sis significant and conclude that some factor other than chance
(our IV) is at work (affecting the DV).
Smith/Davis (c) 2005 Prentice Hall
The Nature and Logic of Probability
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What is considered chance in psychology?
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Typically psychologists say that any event that
occurs by chance 5 times or fewer in 100
occasions is a rare event.
Thus, the common phrase mentioned in
publications is “.05 level of significance.”
Meaning, that a result is considered significant if it
would occur 5 or fewer times by chance in 100
replications of the experiment when the null
hypothesis is actually true.
Smith/Davis (c) 2005 Prentice Hall
A Conceptual Statistical Note
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Although with our hypotheses we discuss comparisons
of sample means, researchers and statisticians want to
compare the two population means represented by the
two groups so they can draw conclusions about the
effects of the IV in the populations of interest.
Thus, the conceptual null hypothesis is μ1 = μ2, and the
conceptual alternative hypothesis is μ1  μ2.
Because we rarely can test populations, we resort to
sampling from those populations and end up
comparing sample means.
Although we use sample means in our formulas, the
statistical null and alternative hypotheses use
Smith/Davis (c) 2005 Prentice Hall
population means.
Probability and the Normal Curve
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The shape and spread of the
normal distribution never changes,
so the probability associated with a
z score of a certain size will never
change.
Thus, to use the normal distribution
to find probabilities, you can use the
z score formula:
This concept is true for statistical
tests in general. We use
probabilities from statistical tests
because they are objective, which
allows us to avoid making
subjective guesses about the
outcomes of our research.
Smith/Davis (c) 2005 Prentice Hall
Probability and Decisions
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The process of using probability to make decisions
concerning experimental findings is fairly
straightforward:
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We use a statistical test to find the probability of a given event
occurring by chance.
We then compare that probability to the critical probability for
making our decisions – the .05 level of significance.
If the probability of the statistic occurring by chance is above
.05, then we decide that chance is still a possible explanation
for the finding, and thus are uncertain about the outcome and
conclude that is possible that the finding is due to chance (i.e.,
accept the null hypothesis).
If the probability of our statistic test is less than .05, we believe
that chance is not a likely explanation for the finding. Thus, the
null hypothesis is rejected and the alternative hypothesis is
accepted.
Smith/Davis (c) 2005 Prentice Hall
Comparing a Sample to a Population:
The One-Sample t Test
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The One-Sample t test – in some
instances we would like to compare a
sample mean to a mean of a
population.
To do such a problem, rather than
using the normal distribution as we did
for the z scores, we need to use a
new statistical distribution: the t
distribution.
The t distribution is similar to the
normal shape, but the t distribution is
used for smaller samples than we
usually would have for the normal
distribution.
The t distribution is flatter in the
middle and higher on the tails
compared to the normal distribution.
Smith/Davis (c) 2005 Prentice Hall
Comparing a Sample to a Population:
The One-Sample t Test
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Being higher on the tails is a
concern because the critical
region for a statistical test
(the .05 level) is located in
the tails of the curves.
Using a normal distribution
for small samples can give
us misleading probabilities
and, therefore, result in
misleading conclusions.
Smith/Davis (c) 2005 Prentice Hall
Comparing a Sample to a Population:
The One-Sample t Test
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Another characteristic of the t distribution that is
different from the normal distribution is that the t
distribution does not retain its shape; it changes
shape based on the size of the sample.
The t distribution changes shape based on its
number of degrees of freedom (i.e., the ability of a
number in a given set to assume any value).
This ability is influence by the restrictions imposed
on the set of numbers. For every restriction, one
number is determine and must assume a fixed or
specified value.
Smith/Davis (c) 2005 Prentice Hall
Comparing a Sample to a Population:
The One-Sample t Test
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When examining a critical value table, you will see
that the values in the df column are continuous until
30, at which point they jump by larger and larger
increments.
Should your value for df not appear in the table, you
should bias the test against yourself.
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That is, never give yourself degrees of freedom that you
don’t actually have.
This decreases the chance of you making a statistical error.
You will also notice in the critical value table, that there are
more probabilities listed than just the .05 level.
Having additional levels in the table allows us to get a better
idea of the probability of chance of our results.
Smith/Davis (c) 2005 Prentice Hall
Comparing a Sample to a Population:
The One-Sample t Test
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The statistical formula for the one-sample t test is:
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Marginal significance is often labeled by the area of
probability between .05 and .10.
The conclusion is that your results were almost
significant, but not quite.
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Typically, researchers will discuss results that are marginally
significant in their articles. However, they will likely use
“hedge words” (e.g., “these results may indicate,” rather
than “these results indicate”) in their discussions.
Smith/Davis (c) 2005 Prentice Hall
One-Tailed and Two-Tailed Tests of
Significance
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You can state your experimental hypotheses in a direction or a nondirectional manner.
Non-directional would look like this: M1  M2
Directional would look like this: either M1 < M2 or M1 > M2
A one-tailed t test evaluates the probability of an outcome in only one
direction (greater than or less than; a directional hypothesis),
whereas the two-tailed t test evaluates the outcome in both possible
directions (a non-directional hypothesis).
For non-directional hypotheses, the probability of the result occurring
by chance alone is split in half and distributed equally in the two tails
of the distribution. Although the test is calculated the same, you would
consult different columns in the t table.
Because the probability is not split for the one-tailed test, the critical
value is lower, and thus it is easier to find a significant result. The
main reason researchers don’t use the one-tailed test is because they
don’t exactly know how an experiment will turn out, and thus are
cautious in their predictions.
Smith/Davis (c) 2005 Prentice Hall
When Statistics Go Astray: Type I and
Type II Errors
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When we conduct research and use probability in determining
significance of our tests, there is always the possibility that your
experiment represents one of those 5 times in 100 when the
results did occur by chance.
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Type I Error – occurs when the null hypothesis is true and you
make an error in accepting the experimental hypothesis. The
experimenter directly controls the probability of making a Type I
error by setting the significance level (e.g., switching from .05 to
.01)
Type II error – occurs when we reject a true experimental
hypothesis. This type of errors is not under the control of the
researcher. We can cut down on Type II errors by implementing
techniques that will cause our groups to differ as much as possible
(e.g., using a strong IV and larger groups of participants are two
techniques that can help avoid Type II errors).
Smith/Davis (c) 2005 Prentice Hall
When Statistics Go Astray: Type I and
Type II Errors
Smith/Davis (c) 2005 Prentice Hall
Sampling Considerations and Basic
Research Strategies
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Sampling
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When we select a group to represent the population we can
that group a sample.
Techniques you can use to obtain a sample include:
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Random sampling - ensures that every member of the
population has an equal chance of being selected for inclusion
in the sample. Thus giving us a representative sample of the
population.
Random sampling without replacement – occurs when a
participant is not eligible to be chosen again once you’ve
selected him/her. This is the technique psychologists prefer.
Random sampling with replacement – occurs when you return
a participant to the population and have him/her eligible for
selection again.
Smith/Davis (c) 2005 Prentice Hall
Sampling Considerations and Basic
Research Strategies
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Sampling
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To increase representativeness of our sample, we
can increase a larger sample size. Generally, the
larger the sample, the more representative it will
be of the population.
Stratified random sampling is another technique
we can use to increase representativeness – it
involves dividing the population into
subpopulations or strata and then drawing a
random sample from one or more of these strata.
Smith/Davis (c) 2005 Prentice Hall
Basic Research Strategies
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Single-strata approach – seeks to acquire data from
a single, specified segment of the population.
Cross-sectional research – involves the comparison
of two or more groups of participants during the
same time, rather limited, time span.
Longitudinal research – involves obtaining a random
sample from the population of interest; then this
sample (or cohort) would be contacted periodically
over an extended period of time to determine if any
changes had occurred during the time of interest.
Smith/Davis (c) 2005 Prentice Hall
Basic Research Strategies
Smith/Davis (c) 2005 Prentice Hall