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T-Statistics, ANOVA and Chi-Square



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Chapter 11 : T-Statistics, ANOVA and Chi-Square



T-Statistics, ANOVA and Chi-Square arrow_upward



T-Test

  • T-test looks at the difference in means of a continuous variable between two groups. 
  • The T distribution is a family of similar probability distributions.
    • A specific T distribution depends on a parameter known as the degrees of freedom.
  • The T statistic allows researchers to use sample data to test hypotheses about an unknown population mean.
    • The advantage of the T statistic is that the T statistic does not require any knowledge of the population standard deviation.
    • The T- Statistic can be used to test hypothesis about a completely unknown population; both  and  are unknown, and the only available information about the population comes from the sample.
  • All that is required for a hypothesis test with T, is a sample and a reasonable hypothesis about the population mean.
  • There are two general situations where this type of hypothesis test is used:

  • ANOVA (Analysis of Variance)

  • ANOVA is used to see an association between a continuous outcome variable and a categorical determining variable.
  • The ANOVA is a statistics option under the means function that allows for testing the difference between the mean outcome scores for the two or more categories of the determining variable.

  • Chi-Square

  • Chi-Square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis.
  • Chi-Square is used to look at the statistical significance of an association between a categorical outcome and a categorical determining variable.

  • Sampling Error and The T-Statistics arrow_upward


  • Whenever a sample is obtained from a population, you expect to find some discrepancy or "sampling error" between the sample mean and the population mean.
  • The goal for a hypothesis test is to evaluate the significance of the observed discrepancy between a sample mean and the population mean.
  • The hypothesis test attempts to decide between the following two alternatives:
    • Is it reasonable that the discrepancy between M and  is simply due to sampling error and not the result of a treatment effect?
    • Is the discrepancy between M and  more than would be expected by sampling error alone? That is, the sample mean significantly different from the population mean?
    • How much difference between M and μ is reasonable to expect?
  • The T-Statistic requires that you use the sample data to compute an estimated standard error of M.

  • Estimated Standard Error arrow_upward


    Where,

  • s = Sample Standard Deviation,
  • n = Number of scores on the test

  • Formula for T-Statistics (one sample test) arrow_upward


  • The one-sample test is used to determine whether the population mean equals a specified value.
  • The T statistic forms a ratio.
  • The top of the ratio contains the obtained difference between the sample mean and the hypothesized population mean.
  • The bottom of the ratio is the standard error which measures how much difference is expected by chance.

  • Formula for Two-Sample Test arrow_upward


  • The two-sample test is used to determine whether the population mean equals a specified value.

  • ANOVA: Analysis of Variance arrow_upward


  • Tests for significant effect of 1 or more factors:
    • Each factor may have 2 or more levels.
    • Can also test for interactions between factors.
    • For just 1 factor with 2 levels, ANOVA = T-test.
  • ANOVA really looks for difference in means between groups (factors & levels).
  • Total variability = Variability due to factors + error.

  • Chi-square Test arrow_upward


  • is used to measure the deviation of observed frequencies from an expected or theoretical distribution.
  • Where,

  • O = Observed frequency (# of events, etc.).
  • E = Expected frequency under H0.


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