Chapter Nine

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Chapter Nine

• Probability, the Normal Curve, and Sampling

PowerPoint Presentation created by Dr. Susan R.
BurnsMorningside College
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The Nature and Logic of Probability

• 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.

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The Nature and Logic of Probability

• 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).

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The Nature and Logic of Probability

• With two groups the null hypothesis looks like
  this
• M1 M2
• 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
• 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.

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The Nature and Logic of Probability

• 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).

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The Nature and Logic of Probability

• What is considered chance in psychology?
• 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.

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A Conceptual Statistical Note

• 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
  population means.

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Probability and the Normal Curve

• 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.

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Probability and Decisions

• The process of using probability to make
  decisions concerning experimental findings is
  fairly straightforward
• 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.

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Comparing a Sample to a Population The
One-Sample t Test

• 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.

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Comparing a Sample to a Population The
One-Sample t Test

• 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.

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Comparing a Sample to a Population The
One-Sample t Test

• 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.

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Comparing a Sample to a Population The
One-Sample t Test

• 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.
• That is, never give yourself degrees of freedom
  that you dont 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.

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Comparing a Sample to a Population The
One-Sample t Test

• The statistical formula for the one-sample t test
  is
• 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.
• 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.

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One-Tailed and Two-Tailed Tests of Significance

• You can state your experimental hypotheses in a
  direction or a non-directional manner.
• Non-directional would look like this M1 ? M2
• Directional would look like this either M1 lt M2
  or M1 gt 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 dont use the
  one-tailed test is because they dont exactly
  know how an experiment will turn out, and thus
  are cautious in their predictions.

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When Statistics Go Astray Type I and Type II
Errors

• 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.
• 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).

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When Statistics Go Astray Type I and Type II
Errors
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Sampling Considerations and Basic Research
Strategies

• Sampling
• When we select a group to represent the
  population we can that group a sample.
• Techniques you can use to obtain a sample
  include
• 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 youve 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.

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Sampling Considerations and Basic Research
Strategies

• Sampling
• 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.

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Basic Research Strategies

• 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.

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Basic Research Strategies