STATISTICAL ANALYSIS.

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STATISTICAL ANALYSIS.

• Your introduction to statistics should not be
  like drinking water from a fire hose!!

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What do you mean by data??
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Statistics 101!!

• Statistics
• Measures of locationmean vs. median and why
• Measures of scalerange, interquartile range,
  standard deviation (and variance)
• Measures of positionpercentiles, deciles,
  quartiles, median
• Note. For categorical variables, we use
  proportions as the descriptive statistics

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Why does lack of normality cause problems?

• When we calculate the p-value for an inference
  test, we find the probability that the sample was
  different due to sampling variability.
  Basically, we are trying to see if a recorded
  value occurred by chance and chance alone. When
  we look for a p-value, we are assuming that all
  samples of the given sample size are normally
  distributed around the mean. This is why the test
  statistic, which is the number of standard
  deviations away from the population mean the
  sample mean is, is able to be used. Therefore,
  without normality, no p-value can be found.

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There are non-parametric tests which are similar
to the parametric tests. The following table
shows how some of the tests match up.
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What is different about Non-Parametric Statistics?

• Sometimes statisticians use what is called
  ordinal data. This data is obtained by taking
  the raw data and giving each sample a rank.
  These ranks are then used to create test
  statistics.
• In parametric statistics, one deals with the
  median rather than the mean. Since a mean can be
  easily influenced by outliers or skewness, and we
  are not assuming normality, a mean no longer
  makes sense. The median is another judge of
  location, which makes more sense in a
  non-parametric test. The median is considered
  the center of a distribution.

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Drawing a histogram..the good the bad and the
downright ugly!!.
Many modern introductory texts and confuse
frequency graphs, relative frequency graphs, and
histograms.
Bad
Good
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What's the difference between a bar chart a
Histogram??
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Critical Values

• For a given number of degrees of freedom, by the
  property of the t-distribution, we know how large
  the t-statistic must be in order to reject the
  null.
• We call that number the critical value of the
  t-statistic and is typically determined by the
  values in a table of the t-statistic.
• If the value of the t-statistic calculated from
  the data is greater than this critical value,
  then we reject the null hypothesis.
• - This is because, for t-statistics greater than
  this critical value, our probability of falsely
  rejecting the null hypothesis is very small.

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Example

• Suppose our null hypothesis is that X is less
  than 0.
• The sample mean is 3
• The sample standard deviation is 2
• There are 121 observations.
• Step 1. We need to establish our critical
  value.
• We wish to reject the null hypothesis if we are
  95 certain that it is false. For 121
  observations and a one-tailed test, the
  critical value is 1.66 (which we look up on the
  table. This corresponds to a significance level
  of .05 with 120 degrees of freedom).
• Step 2. The t-statistic ( 3 0 ) / ( 2 / ?121
  ) ? 3 / .18 ? 16.7.
• Step 3. Compare the t-statistic with the critical
  value. If the t-statistic is greater than the
  critical value, then you can reject the null
  hypothesis.
• In this case, 16.7 is greater than 1.66, so we
  can reject the null hypothesis that X is less
  than zero.

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Example

• The table to the right is a sample cross-tab
• Your research hypothesis is that dog ownership
  and gender are related.
• How do you test this hypothesis?

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Hypothesis Tests about tables

• Step 1. Define null and research hypotheses.
• The null hypothesis will usually be that there
  is no relationship between the rows and the
  columns.
• Step 2. Determine your tolerance for falsely
  rejecting the null hypothesis of no relationship.
• Step 3. Empirically analyse the data to determine
  if there is a relationship.

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Example

• To calculate independence
• 1) Identify the number of respondents in each
  internal cell of the table
• 2) Calculate the number of respondents who would
  be in each cell if independent (corresponds to
  the second number under each total)
• e.g. cell1,1 .5 .15 1000 75
• cell1,2 .5 .85 1000 425
• 3) Compute the chi-squared test statistic (next
  slide)

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The Chi-Square Test Statistic

• To calculate independence
• 3) Compute the chi-squared test statistic
• The chi-squared test statistic is simply
• ??2 ?rows?columns (Observedrow,column -
  Expectedrow,column)2
• Expectedrow,column
• The chi-squared statistic follows a chi-squared
  distribution with degrees of freedom (rows 1)
  (columns 1).

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Example

• If we look at our table of the ??2 with 1 degrees
  of freedom, the critical value for our test
  statistic is 3.84.
• ??2 (100 - 75)2 / 75
• (400-425)2 / 425
• (50- 75)2 / 75
• (450-425)2 / 425
• 19.6
• In this case, we reject the null hypothesis that
  the two populations are statistically independent
  because our test-statistic is greater than our
  critical value.