1. - Interfering factors in the comparison of two sample means using unpaired samples may inflate the pooled estimate of variance of test results. - It is possible to pair the measurements. Only the value of one variable is changed among the two members of each matched pair, but everything else is nearly the same (as closely as possible) for the members of the pairs. -Then the difference between the members of a pair becomes the important variable, which will be examined by a t-test. We test “Is the mean difference significantly different from zero?” Statistical Inference for the Mean: t-test Comparison of paired samples:

2. Statistical Inference for the Mean: t-test Comparison of paired samples: Pair No.123456…n Sample Ax A1 x A2 x A3 x A4 x A5 x A6... x An Sample Bx B1 x B2 x B3 x B4 x B5 x B6 … x Bn d = x Ai - x Bi x A1 - x B1 x A2 - x B2 x A3 - x B3 x A4 - x B4 x A5 - x B5 x A6 - x B6 …x An - x Bn

3. Statistical Inference for the Mean: t-test Comparison of paired samples: Standard Deviation of d: The test statistics t is: Degrees of freedom is n-1 The paired t-test compares the mean difference of pairs to an assumed population mean difference of zero.

4. Test of Significance: Comparing paired samples - State the null hypothesis in terms of the mean difference, such as - State the alternative hypothesis in terms of the same population parameters. - Determine the mean and variance of the difference of the pairs. - Calculate the test statistic t of the observation using the mean and variance of the difference. - Determine the degrees of freedom: - State the level of significance – rejection limit. - If probability falls outside of the rejection limit, we reject the Null Hypothesis, which means the difference of the two samples are significant. Assume the samples are normally distributed and neglect any possibility of interaction. Assume that if the interfering factor is kept constant within each pair, the difference in response will not be affected by the value of the interfering factor. Statistical Inference for the Mean: t-test df = n-1