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AP STATISTICS LESSON 11 – 1 (DAY 3) Matched Pairs t Procedures

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ESSENTIAL QUESTION: How are matched pair designs included in one-sample data analysis? Objectives: To construct t confidence intervals for matched pair designs. To construct t significance tests for matched pair designs. To define Robustness of t procedures.

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Matched Pairs t Procedures The taste test in Example 11.3 was a matched pairs study in which the same 10 tasters rated before-and- after sweetness. Comparative studies are more convincing than single-sample investigations. For that reason, one-sample inference is less common than comparative inference. One common design to compare two treatments makes use of one-sample procedures. In matched pairs design, subjects are matched in pairs and each treatment is given to one subject in each pair.

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Matched Pairs t Procedures (continued…) To compare the responses to the two treatments in a matched pairs design, apply the one-sample t procedures to the observed differences. The parameter μ in a matched pairs t procedure is the mean difference in the responses to the two treatments within matched pairs of subjects in the entire population.

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Example 11.4 Page 629 Floral Scents and Learning The randomization is important because subjects tend to improve their times as they work a maze repeatedly. To analyze these data, subtract the scented time from the unscented time for each subject. The 21 differences form a single sample. Inference procedures for comparing two samples assume that the samples are selected independently of each other. This assumption does not hold when the same subjects are measured twice. The proper analysis depends on the design used to produce the data.

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Example 11.5 Page 632 Is Caffeine Dependence Real? Construct a 90% confidence interval for the mean change in depression score. We are 90% confident that the actual mean difference in depression score for the population is between and points. We estimated that caffeine-dependent individuals would score, on average, between 3.6 and 11.1 points higher on the Beck Depression Inventory when the are given placebo instead of caffeine. This study provides evidence that withholding caffeine from caffeine- dependent individuals may lead to depression.

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Robustness of t Procedures The t confidence interval and test are exactly correct when the distribution of the population is exactly normal. No real data are exactly normal. Because of the tails of normal curves drop off quickly, samples from normal distributions will have very few outliers. Outliers suggest that your data are not a sample from a normal population. Like x and s, the t procedures are strongly influence by outliers.

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Robustness Procedures (continued…) A confidence interval of significance test is called robust if the confidence level or P-value does not change very much when the assumptions of the procedure are violated.

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Example 11.6 Page 635 The Effects of Outliers One of the 46 engines in our sample emitted an unusually high amount (2.94 grams/mile) of NOX. If remove that single data point, What will happen. Fortunately, the t procedures are quite robust against nonnormality of the population when there are no outliers, especially when the distribution is roughly symmetric.

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The Effects of Outliers (continued…) Large samples improve the accuracy of P- values and critical values from the t distribution when the population is not normal. The main reason for this is the central limit theorem. Always make a plot to check for skewness and outliers before you use the one-sample t procedures when N ≥ unless an outlier or quite strong skewness is present.

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Using the t procedure Except in the case of small samples, the assumption that the data are an SRS from the population of interest is more important than the assumption that the population distribution is normal. Sample size at least 15. Use t procedure if the data are close to normal. If the data are clearly nonnormal or if outliers are present, do not use t. Sample size at least 15. The t procedure can be used except in the presence of outliers or strong skewness. Large samples. The t procedures can be used even for clearly skewed distributions when the sample is large, roughly n ≥ 40.

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Example 11.7 Can we use t ? Page 636 and 637 Figure 11.9(a) is a histogram of the percent of each state’s residents who are over 65.We have data on the entire population, we can find μ and σ so we do not confidence intervals or tests. Figure 11.9 (b) Shows the time lightning strike each day in the mountains of Colorado. The data has 70 observations that have a symmetric distribution. You can use t procedures with confidence. Figure 11.9 (c) shows the distribution of word length in Shakespeare ’ s plays is skewed to right. You can use the t procedure for a distribution like this if the sample size is roughly 40 or larger.

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The power of the t test The power of a statistical test measures its ability to detect deviations from the null hypothesis. In practice we carry out the test in the hope of showing that the null hypothesis is false, so higher power is important. The power of the one-sample t test against a specific alternative value of the population mean μ is the probability that the test will reject the null hypothesis when the mean has this alternative value. To calculate the power, we assume a fixed level of significance, usually α = 0.05

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Example 11.8 Page 639 Power of Spanish Teachers.

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