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Two sample t-test: Independent Samples Have data from two samples. In this case from readings of bacteria in carpeted hospital rooms (sample 1) and uncarpeted hospital rooms (sample 2). Want to test that the mean for the uncarpeted is lower than the mean for the carpeted ie H0 : µ1 = µ2 versus HA : µ1 > µ2 . Welcome to Minitab, press F1 for help. MTB > set c1 DATA> 11.8 8.2 7.1 13.0 10.8 10.1 14.6 14.0 DATA> end MTB > set c2 DATA> 12.1 8.3 3.8 7.2 12.0 11.1 10.1 13.7 DATA> end MTB > describe c1 c2 Descriptive Statistics: C1, C2 Variable C1 C2 N 8 8 Variable C1 C2 MTB > MTB > SUBC> SUBC> Mean 11.200 9.79 Minimum 7.100 3.80 Median 11.300 10.60 Maximum 14.600 13.70 help twosample twosample c1 c2; alternative 1; pool. Two-Sample T-Test and CI: C1, C2 Two-sample T for C1 vs C2 C1 N 8 Mean 11.20 StDev 2.68 SE Mean 0.95 1 TrMean 11.200 9.79 Q1 8.675 7.48 StDev 2.677 3.21 Q3 13.750 12.08 SE Mean 0.947 1.13 C2 8 9.79 3.21 1.1 Difference = mu C1 - mu C2 Estimate for difference: 1.41 95% lower bound for difference: -1.19 T-Test of difference = 0 (vs >): T-Value = 0.96 P-Value = 0.178 DF = 14 Both use Pooled StDev = 2.96 MTB > MTB > DATA> DATA> MTB > stack c1 c2 c3 set c4 8(1) 8(2) end boxplot c3*c4 What is the appropriate conclusion? Matched or Paired Data Have data from problem 5 page 278 in Pagano. For each subject we have a reading of LDL cholesterol (mmol/l) from the time they ate cornflakes and the time they ate oat bran. Since one individual generate a pair of observations (observations are matched in pairs), we have matched or paired data. Analysis is done of the difference of the observations. If µd is the population mean of the differences (cornflakes-oatbran), then H0 : µd = 0 and HA : µd > 0. MTB > DATA> DATA> MTB > DATA> DATA> MTB > MTB > set c10 4.61 6.42 5.4 4.54 3.98 3.82 5.01 4.34 3.80 4.56 5.35 3.89 2.25 4.24 end set c11 3.84 5.57 5.85 4.80 3.68 2.96 4.41 3.72 3.49 3.84 5.26 3.73 1.84 4.14 end let c12=c10-c11 desc c12 Descriptive Statistics: C12 2 Variable C12 Variable C12 N 14 Mean 0.363 Minimum -0.450 Median 0.360 Maximum 0.860 TrMean 0.389 Q1 0.098 StDev 0.406 SE Mean 0.108 Q3 0.732 MTB > ttest c12; SUBC> alternative 1. One-Sample T: C12 Test of mu = 0 vs mu > 0 Variable C12 Variable C12 N 14 Mean 0.363 StDev 0.406 95.0% Lower Bound 0.171 T 3.34 SE Mean 0.108 P 0.003 What is the conclusion? Power and Sample Size Calculations In carrying out a test of hypothesis, we often want to choose a sample size to give us certain characteristics of the test. Alternatively we may have a fixed sample size and want to determine the characteristics of the test. Typically we fix the size of the test which is the probability of making a type one error, usually denoted by α. For a fixed value in HA , the power is the probability of rejecting HA under the value of the parameter specified. Power is 1-probabilty of making a type II error, so we want the power to be large. The steps for computing the power are as follows: 1. Specify α, often 0.05. 2. Specify the difference we want to detect ie if we are comparing two means, we would specify how far apart the means have to be to be of practical importance. 3. Give the sample size and standard deviation, sigma. 3 4. Run Minitab command tone or ttwo with subcomands alpha=,sigma=,difference=, sample=. Answer is the power. In order to determine the sample size needed to give a power of a certain value, the steps are as follows: 1. Specify α, often 0.05. 2. Specify the difference we want to detect ie if we are comparing two means, we would specify how far apart the means have to be to be of practical importance. 3. Give the desired power and standard deviation, sigma. 4. Run Minitab command tone or ttwo with subcomands alpha=,sigma=,difference=, power=. Answer is the sample size required. Examples of power calculations based on the hospital rooms example. First we compute the power for detecting a difference of 1.4 (the observed difference). We then ask what sample size we need to get a power of 0.8 for this difference. MTB > SUBC> SUBC> SUBC> SUBC> SUBC> power; ttwo; alpha .05; diff 1.4; sigma 3; sample 8. Testing mean 1 = mean 2 (versus not =) Calculating power for mean 1 = mean 2 + difference Alpha = 0.05 Sigma = 3 Difference 1.4 Sample Size 8 Power 0.1403 MTB > power; SUBC> ttwo; 4 SUBC> SUBC> SUBC> SUBC> alpha .05; diff 1.4; sigma 3; power .8. Power and Sample Size 2-Sample t Test Testing mean 1 = mean 2 (versus not =) Calculating power for mean 1 = mean 2 + difference Alpha = 0.05 Sigma = 3 Difference 1.4 Sample Size 74 Target Power 0.8000 Actual Power 0.8051 5