Chapters 22, 24, 25 Inference for Two-Samples

Confidence Intervals for 2 Proportions

Categorical Data We use the 2 Proportion Z Interval when we have two independent samples from categorical data and we want an interval to estimate the true difference in those proportions..

Comparing Two Proportions Conditions: *Randomness for both samples *Independence within both samples *n 1 p 1, n 1 (1-p 1 ), n 2 p 2, n 2 (1-p 2 ) are all  10 (insures both samples are large enough to approximate normal) *Independence between both samples

Pooling Pooling means we are combining our sample sizes to reduce variability. We can only pool under certain conditions. The formula chart reminds you when you are allowed to pool.

For Confidence Intervals: (we do not assume that p 1 = p 2 therefore we do not pool for CI's) * choose the non-pooled formula from chart, calculator will make correct choice automatically

For Confidence Intervals:  Our confidence interval statements reflect the true difference in the two proportions (in context).  i.e. I am 90% confidence that the true difference in ….

Comparing Two Means  When our data is quantitative, then we are either looking for an interval that contains the true difference in the means or the true mean difference.

Conditions: *Randomness for both samples *Independence within both samples. *Sample size restriction must be met: If n 1 + n 2 < 15, do not use if outliers or severe skewness are present If 15 ≤ n 1 + n 2 < 30, use except in presence of outliers If n 1 + n 2  30, sample is large enough to use regardless of outliers or skewness by CLT * The two samples we are comparing must be independent from each other.

Comparing Two Independent Means  Two-Sample t Interval:(Quantitative Data) * choose the non-pooled formula from the formula sheet *df = there is a nasty formula – ick! But …calculator gives you this

 When we interpret confidence intervals for independent samples we say “the true difference between two means” (not the mean difference).  i.e. We are 95% confident that the true difference in the mean ….

Paired Samples  Often we have samples of data that are drawn from populations that are not independent. We have to watch carefully for those!! We can not treat these samples as two independent samples, we must consider the fact that they are related.  So…what do we do?

Paired Samples  If we have two samples of data that are drawn from populations that are not independent, we use that data to create a list of differences. That list of differences then becomes our data and we will not use the two individual lists of data again.

Paired Samples  Once we have the list of differences, everything else is like 1 sample procedures from the last unit.

Paired Samples  We will treat that list of differences as our data.  That list must meet the conditions for a single sample t-distribution. We will use the t-interval or the t-test on this list of differences depending on the question.  When we interpret our confidence interval or make our conclusion we will be talking about the “mean of the differences.”

The One-Sample t-Procedure:  A level C confidence interval for  is: where t* is the critical value from the t distribution based on degrees of freedom.

Just remember that all the variables represent the “mean difference” for your populations.

Hypothesis Testing with Two Samples

Categorical Data Ho: p 1 = p 2 Ha:p 1  p 2 or p 1 < p 2 or p 1 > p 2 (Since our null states that p 1 = p 2, we assume this to be true until proven otherwise, therefore we automatically pool here)

Where *choose the pooled formula from the chart, the calculator will automatically pool We rely on the same p-values and alphas to make our conclusions.

Quantitative Data Ho: μ 1 = μ 2 Ha: μ 1  μ 2 or μ 1 < μ 2 or μ 1 > μ 2  Since there is no assumption of σ 1= σ 2, we do not pool

Two-Sample t-test *choose the non-pooled formula from the formula sheet *df = there is a nasty formula – ick! But …calculator gives you this

Two-Sample t Procedures:  We do not pool t-tests.  Pooling assuming equal σ values. Since σ 1 and σ 2 are both unknown, why would we assume they are equal?  You must tell the calculator you do not want to pool. “Just say No”.

Paired Samples  Remember when your two samples are not independent of each other, you must create a list of differences and use that set of data and single sample procedures.

Single Sample Reminder  To test the hypothesis: Ho:  =  o and Ha:  >  o Ha:  <  o Ha:    o  Calculate the test statistics t and the p value.  We make the same conclusions based on p-values and alpha.

Single Sample Reminder  For a one-sample t statistic: has the t distribution with n – 1 degrees of freedom.  Just remember that all the variables represent the “mean difference” for your populations.