- Comparing one sample average/percentage to an "external" standard (a number given to us by the claim we are testing)

- Example: A sample of cereal boxes contain, on average, 2.95 oz of raisins, while it is claimed (a number given to us) that the average of all boxes is 3.0 oz.
- Use one-sample z test (assuming the sample size is over 25-30)
- For small samples, use one-sample t test
- This t-test assumes that the population is approximately normally distributed, and we are estimating the population S.D. from the sample.
- Comparing sample average (or percentages) of 2 samples (no external standard or number given)
- Example: A sample of freshman shows 20% own a refrigerator, while a sample of seniors shows 33% ownership. Is this difference chance variation in our sample, or do seniors (all seniors, overall) own refrigerators at a higher rate than freshmen (all freshmen, overall)?
- Use two-sample z test (assuming the sample sizes are over 25-30)
- For small sample sizes, use two-sample t test (we did not study this test)
- Either test assumes the samples are independent and small compared to their respective populations (so that non-replacement isn't an issue).
- Taking an entire group and dividing it in two at random to compare the results of treatment vs. placebo (or to compare two different treatments)
- Example: Testing the effectiveness of Vitamin C in treating colds
- Use two-sample z test, assuming the groups are over 25-30 in size.
- The fact that the two samples are not independent, i.e. that the choice of the first sample determines the second sample, causes one type of error, and...
- The fact that the sample is large relative to the overall population, i.e. half of the overall population, causes another type of error, and...
- These two types of errors cancel each other out.

- Chi-squared tests: working with qualitative variables

- When an external standard is given, e.g. testing dice for
fairness - we are given the expected percentage for each outcome by the null hypothesis.

- When we are testing two qualitative variables for independence
- When the two variables both have only two values, this can be
recast as a two-sample z or t test; one will get approximately the same P-value, hence the same conclusion.

- Analysis of Variance (ANOVA) (we have not studied this test)

- For testing the independence of a qualitative variable from a quantitative variable, e.g. is GPA independent of residence hall?
- When the qualitative variable has only two values, this can be recast as a two-sample z or t test; one will get approximately the same P-value, hence the same conclusion.

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