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

### Topics covered in this snack-sized chapter:

#### 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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