6.3 Two-Sample Inference for Means November 17, 2003

Paired Differences Matched Pairs Design experimental plan where the experimental units are divided into halves and two treatments are randomly assigned to the halves Attempting to determine if there is a significant difference between the mean responses of the treatments

Procedure Obtain the difference between responses for each experimental unit Analyze the differences using a one- sample approach If a large sample is obtained, use critical values from the Standard Normal distribution (z) Otherwise, use critical values from the corresponding t distribution

Example 9 pg 370 Students worked with a company on the monitoring of the operation of an end-cut router in the manufacture of a wood product. They measured the critical dimensions of a number of pieces of a type as they came off the router. Both a leading- edge and a trailing-edge measurement were made on each piece. Both were to have a target value of.172 in.

PieceLeading Edge Trailing Edge

PieceLeading Edge Trailing Edge Difference

Confidence Interval

Hypothesis Test Is there a significant difference between the measurements at α =.01? Ho: µ = 0 Ha: µ ≠ 0

Independent Samples The goal of this type of inference is to compare the mean response of two variables (or treatments) when the data are not paired or matched A key assumption that will be made is that the separate samples used to collect information concerning the two variables are independent One sample does not influence the other sample in any way Furthermore, one must assume that both sampled populations are normally distributed

Large Sample Comparison The quantity of interest is a linear combination of population means, namely µ 1 - µ 2 The above quantity will be estimated by As a result, various quantities of the sampling distribution of the difference, under the assumption of equality between the means, need to be developed Ho: µ 1 - µ 2 = # Ha: µ 1 - µ 2 ≠ #

Test Statistic

Confidence Interval

Example A company research effort involved finding a workable geometry for molded pieces of a solid. A comparison was made between the weight of molded pieces and the weight of irregularly shaped pieces that could be poured into the same container. A series of 30 attempts to pack both the molded and the irregular pieces of the solid were compared. Is there enough evidence to suggest that the irregular pieces produced higher weights?

The Data 1 = molded n 1 = 30 s 1 = 9.31 = =irregular n 2 = 30 s 2 = 8.51 =

Small Samples If at least one sample size is small, then use critical values from a t distribution for constructing confidence intervals and performing hypothesis tests with degrees of freedom obtained by Satterthwaite’s Approximation

Satterthwaite’s Approximation

Test Statistic

Confidence Interval

Example The data shown gives spring lifetimes under two different levels of stress (900 and 950 N/mm 2 ). Do the data give evidence of a significant difference at α =.05? 950 Level900 Level

Minitab Output Descriptive Statistics: 950, 900 Variable N Mean Median TrMean StDev SE Mean

Assignment Page 385: #3, #4