Statistics Chapter 9: Two samples

Required Conditions for Two Large Samples (population mean)

1. The two samples are randomly and independently selected from the target populations

2. The sample sizes are both >= 30

Interpretation of CI

• “Using this estimation procedure over and over again for different samples, we know that approximately 100(1-α)% of the confidence intervals in this manner will enclose the μ1 - μ2”

• “Therefore, we are 100(1-α)% confident that the mean …. lies somewhere between (x,y)”

Required Conditions for Two Small Samples (population mean)

1. The two samples are randomly selected in an independent manner from the two target populations

2. Both sampled populations have distributions that are approximately normal

3. The population variances are equal, but unknown

4. Sample sizes are small, to check: ½ <= S1/S2 <= 2

Decision and Conclusion of Hypotheses

• If the calculated value of the proposed test statistic belongs to the rejection region, we reject Ho, otherwise we fail to reject Ho

• “We can conclude that there is sufficient/insufficient evidence at α% level of significance to support/ reject the claim that…”

Paired samples

the two samples are not independent, Y1 and Y2 are linked in some way, usually by a direct relationship

Paired data

• Calculate the differences between each pair Xd = X1 - X2

• Paired t-tests use this difference data to calculate the mean of the differences and the standard error of the differences

Paired Difference CI

“Let μ be the mean difference of population differences between Population I and Population II”

Required Conditions for Two Paired Samples

1. A random sample of differences is selected from the target population differences

2. The population of differences has a distribution that is approximately normal

Hypotheses Test for Paired Difference

Label de parameters:

• μ1 = the population mean for Population I

• μ2 = the population mean for Population I

Test: “We will use small single-sample t-test for paired data”