9-1 Two Proportions

• When testing a claim about two population proportions, the p-value method and the critical value method are equivalent, and the confidence interval is NOT equivalent to the p-value method or the critical value method

• For tests of hypotheses made about 2 population proportions, we consider only tests having a null hypothesis of p1 = p2

• Do not test for equality of two population proportions by determining whether there is an overlap between two individual CI estimates of the two individual population proportions

• If the requirement that we have 2 simple random samples is violated, there is probably nothing that can be done to salvage them

• If the requirement that each of the 2 samples have at least 5 successes and at least 5 failures in a hypothesis test, we can use Fisher's exact test to provide an exact p-value instead of using the method based on a normal distribution approximation

• If the requirement that each of the 2 samples have at least 5 successes and at least 5 failures in a confidence interval, we can use bootstrap resampling methods to construct a confidence interval

9-2 Two Means: Independent Samples

• Two sample are independent if the sample values from one population are not related to or somehow naturally paired or matched with the sample values from the other population

• Two samples are dependent (or consist of matched pairs) if the sample values are somehow matched, where the matching is based on some inherent relationship

• If the two samples have different sample sizes with no missing data, they must be independent. If the two samples have the same sample size, the samples may or may not be independent.

• The p-value method of hypothesis testing, the critical value method of hypothesis testing, and confidence intervals all use the same distribution and standard error, so they are all equivalent in the sense that they result in the same conclusions

9-3 Two Dependent Samples (Matched Pairs)

• When designing an experiment or planning an observational study, using dependent samples with matched pairs is generally better than using two independent samples

• Procedures for inferences with dependent samples:

  • Verify the sample data consists of dependent samples

  • Find the difference d for each pair of sample values

  • Find the value of d bar (mean of the differences) and s subscript d (standard deviation of the differences)

  • For hypothesis tests and CI, use the same t test procedures used for a single population mean

• Alternative method used when population is not normal and when n <= 30: bootstrap

9-4 Two Variances or Standard Deviations

• Utilize the F test for testing claims made about two population variances or standard deviations

• If the two populations have equal variances, then the ratio s1 squared / s2 squared will tend to be close to 1

• Large values of F are evidence AGAINST sigma 1 squared = sigma 2 squared

• The count five method is a relatively simple alternative to the F test, and it does not require normally distributed populations

• The Levene-Brown-Forsythe test is another alternative to the F test, and it is much more robust against departures from normality