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Inferential Statistics


Inferential Statistics Statistical Analysis of Research Data * * * * * * * * * * * * * * * * * * * * * * * * * Statistical Inference Getting information about a ... – PowerPoint PPT presentation

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Title: Inferential Statistics

Inferential Statistics
  • Statistical Analysis of Research Data

Statistical Inference
  • Getting information about a population from a
  • How practical are statistically significant
  • Cost/benefit
  • Crucial difference
  • Client acceptability
  • Public and political acceptability
  • Ethical and legal concerns

Inferential Statistics
  • These provide a means for drawing conclusions
    about a population given the data actually
    obtained from the sample. They allow a
    researcher to make generalizations to a larger
    number of individuals, based on information from
    a limited number of subjects. They are based on
  • Probability theory
  • Statistical inference
  • Sampling distributions

Inferential Statistics
  • Probability theory the basis for
    decision-making statistical inferences. It refers
    to a large number of experiences, events or
    outcomes that will happen in a population in the
    long run. Likelihood and chance are similar
    terms. Examples are usually based on tossing a
    coin and finding heads or tails. Probabilities
    are statements of likelihood expressed in values
    from 0 to 1.0.
  • p the number of outcomes
  • the total possible outcomes

Inferential Statistics
  • Statistical inference statistics enable us to
    judge the probability that our inferences or
    estimates are close to the truth
  • Sampling distributions are theoretical
    distributions developed by mathematicians to
    organize statistical outcomes from various sample
    sizes so that we can determine the probability of
    something happening by chance in the population
    from which the sample was drawn. They allow us
    to know the relative frequency or probability of
    occurrence of any value in the distribution.

Inferential Statistics
  • Hypothesis testing 5 basic steps
  • Make a prediction
  • Decide on a statistical test to use
  • Select a significance level and a critical region
    (region of rejection of the null hypothesis). To
    do this you must consider two things
  • Whether both ends (tails) of the distribution
    should be included.
  • How the critical region of a certain size will
    contribute to Type I or Type II errors.

Critical Region in the Sampling Distribution for
a One-Tailed Test
Critical Regions in the Sampling Distribution for
a Two-Tailed Test
Levels of Significance
  • Remember, if a printsout shows a two-tailed test
    result, and you wanted a one-tailed result,
    divide the two tailed p value by 2.
  • Example p .080 (two-tailed) or pgt.05
  • p .040 (one-tailed) or plt.05
  • The first would not be statistically significant,
    whereas, the second would be statistically

Outcomes of Statistical Decision Making
Inferential Statistics
  • Hypothesis testing cont.
  • Computing the test statistic - The test statistic
    is not a mean, sd or any form of descriptive
    data. It is simply a number that can be compared
    with a set of results predicted by the sampling
  • Compare the test statistic to the sampling
    distribution (table) and make a decision about
    the null hypothesis reject it if the statistic
    falls in the region of rejection.
  • Consider the power of the test its probability
    of detecting a significant difference
    parametric tests are more powerful

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Degrees of Freedom
  • One sample t-test or paired t-test N-1
  • Independent t-test N-2
  • Chi-square test
  • ( rows - 1) x ( columns 1)
  • df between groups ( levels or groups
  • df within groups ( subjects - of
  • Correlations N-2

Levels of Measurement
  • There are four levels or scales of measurement,
    Each level is classified according to certain
    characteristics. Data that fall in the first
    level are limited to certain statistical tests.
    Choices of statistical tests (and the power of
    the tests) increase as the levels go up.

Levels of Measurement cont.
  • Nominal scale measurement at its weakest
    numbers or other symbols are used to classify or
    partition a class into mutually exclusive
    subclasses animals can be classified as dogs,
    cats, etc. You can test hypotheses regarding
    distribution among the categories by using the
    Chi-square test.

Levels of Measurement cont.
  • Ordinal scale shows relationships among
    classes, such as higher than, more difficult
    than, etc. It allows the attributes of a variable
    to be ranked in relation to each other. A
    researcher can test hypotheses using
    non-parametric statistics of order and ranking.

Levels of Measurement cont.
  • Interval scale is similar to the ordinal scale,
    but the distance between any two numbers is of a
    known size. The numbers used have absolute values
    and the interval between each number is
    considered to be equal. Increasing amounts of a
    variable are represented by increasing numbers on
    the scale. The variable is continuous. There is
    no true zero where you have none of the
    variable. All parametric tests can be used with
    interval data.

Levels of Measurement cont.
  • Ratio scale is like the interval scale but it
    has a true zero point as its origin. Time,
    length and weight are ratio scales when used
    alone, but not as a characteristic of a person.
    Arithmetic, all parametric tests and geometric
    means can be used with ratio data.

Tests of Significance
  • Parametric tests of significance used if there
    are at least 30 observations, the population can
    be assumed to be normally distributed, variables
    are at least in an interval scale
  • Z tests are used with samples over 30. There are
    four kinds (two samples or two categories)
  • t-tests are used when samples are 30 or less.
  • Single sample t-test (one sample)
  • Independent t-test (two samples)
  • Paired t-test (two categories

Tests of Significance
  • Non-parametric tests of significance small
    numbers, cant assume a normal distribution, or
    measurement not interval
  • Chi-square requires only nominal data allows
    researcher to determine whether frequencies that
    have been obtained in research differ from those
    that would have been expected use a X2 sampling
  • Chi-square goodness of Fit
  • Chi-Square test of independence

Tests of Significance
  • Mann Whitney U an alternate to the independent
    t-test must have at least ordinal data. It
    counts the comparative ranks of scores in two
    samples (from highest to lowest) The null
    hypothesis is that the two samples are randomly
    distributed. Use U sampling distribution tables
    for small sample sizes (1-8) and medium sample
    sizes (9-20) and the Z test for large samples

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Tests of Significance
  • Wilcoxin Matched Pairs (signed rank test) is an
    alternate to the paired t-test. It is used for
    repeated measures on the same individual. It
    requires a measurement between ordinal and
    interval scales the scores must have some real
    meaning. Use a T table. If the T is less than or
    equal to the T in the table, the null hypothesis
    is rejected.

Measures of Association
  • Parametric Measures of Association These answer
    the question, within a given population, is
    there a relationship between one variable and
    another variable? A measure of association can
    exist only if data can be logically paired. It
    can be tested for significance.
  • Correlation answers the question, What is the
    degree of relationship between x and y Use
    Pearson Product Moment Correlation (Pearson r )
    see next slide

Measures of Association The Pearson Correlation
Coefficient (Pearson r)
  • The r examines the relationship between two
    quantitative sets of scores.
  • The r varies from 1.00 to 1.00
  • The r is not a proportion and cannot be
    multiplied by 100 to get a percentage.
  • To think of the r as a percentage, it needs to be
    converted to the Coefficient of Determination
    or R2 . An r of .50 is 25 better than an r of

Measures of Association
  • Non-parametric tests for association
  • Correlation
  • The Spearman Rank Order Correlation (Rs) To
    what extent and how strongly are two variables
  • Phi coefficient it can be used with nominal
    data, but should have ordinal data
  • Kendalls Q can be used with nominal data

  • Parametric Prediction using a correlation, if
    you know score x, you can predict score y for
    one person Use regression analysis
  • Simple linear regression allows the prediction
    from one variable to another you must have at
    least interval level data
  • Multiple linear regression this allows the
    prediction of one variable from several other
    variables. The dependent variable must be on the
    interval scale

  • Non-parametric Prediction measures the extent
    to which you can reduce the error in predicting
    the dependent variable as a consequence of having
    some knowledge of the independent variable such
    as, predicting income DV by education IV
  • Kendalls Tau used with ordinal data and
    ranking - is better than the Gamma because it
    takes ties into account
  • Gamma - used with ordinal data to predict the
    rank of one variable by knowing rank on another
  • Lambda can be used with nominal data
    knowledge of the IV allows one to make a better
    prediction of the DV than if you had no knowledge
    at all

Parametric Multiple Comparisons
  • The analysis of variance (ANOVA) is probably the
    most commonly encountered multiple comparison
    test. It compares observed values with expected
    values in trying to discover whether the means of
    several populations are equal. It compares two
    estimates of the population variance. One
    estimate is based on variance within each sample
    within groups. The other is based on variation
    across samples between groups. The between
    group variance is the explained variance (due to
    the treatment) and the variation within each
    group is the unexplained variance (the error

Parametric Multiple Comparisons
  • ANOVA cont. The ratio of the explained scores to
    the unexplained scores gives the F statistic. If
    the variance between the groups is larger, giving
    an F ratio greater than 1, it may be significant
    depending upon the degrees of freedom. If the F
    ratio is approximately 1, it means that the null
    hypothesis is supported and there was no
    significant difference between the groups.

Parametric Multiple Comparisons
  • ANOVA cont. If the null hypothesis is rejected,
    then one would be interested in determining which
    groups showed a significant difference. The best
    way to check this is to conduct a post hoc test
    such as the Tukey, Bonferrioni, or Scheffe. (SPSS
    will do this for you if you click on Post-hoc and
    check the test desired.Check on descriptives
    while you still in ANOVA, and the program will
    also give you the mean for each group)

Parametric Multiple Comparisons
  • Two-Way Analysis of Variance
  • Classifies participants in two-ways
  • Results answer three questions
  • Two main effects
  • An interaction effect

Non-parametric Multiple Comparison
  • Kruskal-Wallis Test an alternative to the
    one-way ANOVA. The scores are ranked and the
    analyses compare the mean rank in each group. It
    determines if there is a difference between
  • McNemar Test an adaptation of the Chi-square
    that is used with repeated measures at the
    nominal level.
  • Friedman Test an alternative to the repeated
    ANOVA. Two or more measurements are taken from
    the same subjects. It answers the questions as to
    whether the measurement changes over time.