Student’s t-test
POLS 3316: Statistics for Political Scientists

Hypothesis tests

\(X^2\) and t-test are simple enough to work out with a calculator, so a small data version of each will be on the final

• For \(X^2\), I may give an intermediate step such as the column and row totals (marginal frequencies) and expected values and let you solve from there.
• I might also give you the t-score and ask you to find the p-value from a table.
• There will be at least one question where you work through one of these tests from start to finish.

Chi-Square review

• \(X^2\) (Chi-squared) test
• For categorical variables
• aka as cross-tabs because of the format
• worked through a \(X^2\) test problem together
• This is on Problem Sets 5/6 and final

Today

• Concepts of Z-score (review), Student’s t-test, and ANOVA
• Where each one is appropriate
• Brief discussion of the formulas for all three
• We will work through a paired sample t-test together as our second example of hypothesis testing
• Paired t-test is on Problem Set 5/6 and final

Hypothesis test uses

• \(X^2\) (Chi squared): Categorical variables with counts
• Student’s t-test: compares the means of two groups
• z-score: continuous, normally distributed variables
• ANOVA (Analysis of Variance)
• Lots of others!

Hypothesis test uses

• \(X^2\) (Chi squared): Categorical variables with counts

• \(X^2\) test of goodness of fit - tells whether the sample data is representative of the population

• \(X^2\) test of indendependence (we dit this) - tells us if two categorical variables are related or not

Student’s t-test: compares the means of two groups

Such as the sugar content of barley malt for use in brewing beet…

https://www.youtube.com/watch?v=U9Wr7VEPGXA

Student’s t-test: compares the means of two groups

• Developed by William Sealy Gosset, who published under the pseudonym Student
• Gosset worked for Guinness and was interested in the quality of barley malt for use in brewing beer
• He was interested in small sample sizes, so he developed the t-test
• He also developed the concept of statistical power
• He was a chemist, not a statistician, so he published under a pseudonym

Student’s t-test: compares the means of two groups

• pairwise comparison: what are the pairs?

    - one sample: comparing one group against a standard value
    - two-sample or independent t-test: compares two groups from different populations 
    - paired t-test: compares a single group as in before and after comparison

• One or two tails

    - Two tailed test: tells if they are different, either greater or less
    - One tailed test: tells if one group is specifically greater or less, bot not either

Other points:

    - degrees of freedom = n - 1
    - When t-test degrees of freedom > 30, it converges on the z-score
    - t-test is more conservative than z-score
    

More conservative than the z-score: distribution of t-scores

t-distribution

More conservative than the z-score: distribution of t-scores

t vs z dist

z-score: continuous, normally distributed variables

• Continuous variables

• normal distribution

    - Central Limit Theorem can get us to normal distribution

• known population standard deviation

    - "known" ~ accepted estimate of the population standard deviation from LLN and CLT

• Use if: if the population standard deviation is known or reliably estimated and sample size > 30

ANOVA (Analysis of Variance)

• tests difference of means between 3 or more indendepnt groups
• This is often used to test removing variables one at a time from a multi-variable model (something we will not be doing)
• Uses the F-test
• The same F-test in regression results - model fit

Deeper look at t-tests

• Paired sample t-test
• \(t = \frac{\bar{x}_{diff}}{\sigma_{diff}/\sqrt{n}}\)

• \(\bar{x}_{diff}\): sample mean of the differences
• \(\sigma_{diff}\): sample standard deviation of the differences
• n: sample size (in pairs)

Short example

Second Example Data:

Group 1: (12.2, 14.6, 13.4, 11.2, 12.7, 10.4, 15.8, 13.9, 9.5, 14.2) Group 2: (13.5, 15.2, 13.6, 12.8, 13.7, 11.3, 16.5, 13.4, 8.7, 14.6)

More on reading t-tables plus 1- and 2- tailed tables here:

https://www.statisticshowto.com/tables/t-distribution-table/

Authorship, License, Credits

• Author: Tom Hanna

• Website: tomhanna.me

• License: This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.