Inferential statistics

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Inferential statistics

• Hypothesis testing

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The goals of quantitative research

• We usually want to make some statements about how
  things are related
• Does Sesame Street teach kids to read?
• Does watching TV news lead to support for the
  Iraq War?
• Do young men enjoy horror films more than young
  women do?

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Studying relationships quantitatively

• We use statistics to draw conclusions from
  quantitative data
• To look for relationships among variables, we
  want to determine
• Whether a relationship exists in the sample data
• Whether that relationship can be generalized to
  the population we sampled from

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Does a population characteristic differ from some
expectation?

• For example, do Telecommunications students
  (population) spend more than 5 hours a week
  playing video games?
• Take a sample of Tel students and ask them how
  many hours per week they play video games.
• If the mean is higher than 5, you would go on to
  see whether the difference you found was
  statistically significant.

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Differences between two groups

• You may want to see if two groups are different
  on some characteristic
• If the difference between the two groups is large
  compared to the variance within the groups, then
  there will be a statistically significant
  difference between them

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For example

• Do male students spend more time watching sci-fi
  than do female students?
• Ask a sample of students, some male and some
  female, how many hours they spend watching sci-fi
  per week
• Determine the mean number of hours male and
  female students watch Sci-Fi

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To test for a difference

• If the mean number of sci-fi viewing hours
  differs quite a bit between males and females
  relative to the variation among males and among
  females, then you would conclude that there is a
  real difference within the population (not just
  in your sample)
• You accept your sample estimate of the difference
  between males and females as the best estimate of
  the population difference
• NOTE There are statistics to help make this
  judgment

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Females
Males
of viewers
Hours viewing
Mean scifi viewing hours among
females
Mean scifi viewing hours among males
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Statistics for comparisons among groups

• t-test
• preferred where two groups can be compared based
  on some hypothesis
• ANOVA
• comparison among multiple groups
• allows for factorial designs
• good at dealing with interactions
• These are the main statistics for analyzing
  experimental data

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Example Effects of gender and commercial message

• We need to choose between cell phone commercials
• We want to know which commercial to use, whether
  men or women spend more on cell phones, and
  whether the commercials have different effects by
  gender

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Main effect of gender
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Main effect of commercial
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Interaction
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Relationships among variables

• When both your variables can take many values,
  you may want to generate a scatterplot
• Values on one variable are represented on the X
  axis
• Values on the other variable are represented on
  the Y axis

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• The relationship between the variables is
  estimated by fitting a line to the data
• The line is fitted in a particular way that
  provides the least total error or distance
  between the points and the line

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Measures of association

• If we want to know to what extent two variables
  are related, we look at a measure of association

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• Different measures are used depending upon
  whether the data were collected using nominal,
  ordinal, interval or ratio scale
• Parametric
• Nonparametric
• Two nominalChi-square (non-parametric)
• Two ordinalSpearmans rho
• Two intervalPearsons r

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Chi-square

• Because the data provide no direction or distance
  on the scales they use, Chi-square is based on
  the percentage of subjects found in each cell of
  a contingency table

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Covariance among variables

• Correlation
• Pearsons r
• r2 coefficient of determination
• how much of variance in one variable can be
  accounted for by variance in another

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Linear regression

• Minimizes the total squared distances between
  individual data points and a constructed
  regression line
• yaxb
• allows for prediction of the behavior of the
  dependent variable
• preferred to simple correlation

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The problem with outliers

• Extreme cases (outliers) can unduly influence
  measures of association

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Multiple variables

• We can statistically analyze the relations among
  multiple variables at the same time
• Multiple correlation
• Multiple regression
• Control for multiple variables
• Unique contribution of a single variable

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Statistical significance

• Because you have sampled from a population, your
  results may have occurred largely by chance
• Researchers usually want to be certain that their
  findings are unlikely to be a result of chance
• But you cant entirely eliminate chance, so you
  set a limit on how much chance you are willing to
  accept

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• So, you set a p level
• .05
• Fewer than than 5 samples in 100 would generate a
  result this unlikely
• 95 in 100 samples would generate a mean estimate
  within a designated distance of the sample
  estimate
• If your findings meet this criterion, they are
  statistically significant
• Other common significance levels are .01 or .001

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• Statistically significant findings are not
  necessarily theoretically significant
• If you have a large sample, a relatively small
  effect size may be statistically significant

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So

• We often want to determine whether variables are
  related in a population
• We use appropriate sample statistics to determine
  whether they are related in our sample
• We use inferential statistics to evaluate whether
  we think what we found in our sample is true of
  the larger population