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Comparing Means One-sample t-test Independent-sample t-test Paired-sample t-test One-way ANOVA By Qin Xiaoqing Normal distribution and sampling distribution Normal distribution is made up of individual scores. Sampling distribution is composed of group means. With more groups means, a curve of their distribution would be symmetric. This symmetric curve is not called a normal distribution but rather a sampling distribution of means A sampling distribution is the distribution of a sample statistic that would be obtained if all possible samples of the same size were drawn from a given population. Means and standard deviation for sampling distribution When computing a mean, the individual differences are averaged out. The high and low scores disappear and only a central measure (mean) was left. Since sampling distribution is made of means, it is much more compact then scores in a single distribution. Therefore, the standard deviation will be smaller in the distribution of means. Population mean and standard error When taking the average of a group of scores, we call the central balance point the mean. When taking the average of a group of means, we call it the population mean. The standard deviation of the sampling distribution is called the “standard error” and, when based on sample statistics, it estimates random sampling error. As the size of the sample increases, sampling error or the standard error will decrease. Sample size and distribution The size of the groups will also influence the sampling distribution of means. The larger the N for each group, the more the means will resemble each other. The more scores there are in each group, the greater the chance of a normal distribution. With a normal distribution, the mean becomes quite precise as a measure of central tendency. The means of large classes should be very similar to each other. Features of sampling distribution For 30 or more samples, sampling distribution of means is normally distributed. Its mean is equal to the mean of the population. Its standard deviation, called standard error of means, is equal to the standard deviation of the population divided by the square root of the sample size. From sample to population Sample Statistic Population Estimates Parameter Comparison between the mean of sample and that of population x μ Zx Sx Where, x is the mean of sample, μis the population mean, and S is the standard deviation of means. x Sx Sx N Where Sx is the standard deviation of the sample, N is the size of the sample. One-sample t-test H0=There is no effect of group on the dependent variable. If the mean for the class was 80, the published μfor the test was 65, SD for the class was 30, the class size was 36. T value = 3.0 Df = N-1 = 36-1 =35 The t critical value for df 30= 2.042 (p<.05), 2.750 (p<.01), t obs x μ Sx 80 65 t obs Sx 80 65 t obs 30 36 Exercise for one-sample test 14 students, mean of reading achievement for the class is 80 and SD is 5, the published mean is 85. It is believed that all the students were better than average. Research hypothesis? Significant level? 1- or 2-tailed? Dependent variable? Independent variable? t-value? Critical value? df? Key to the exercise 80 85 t obs 3.73 5 14 t critical value (p.05) = 2.16 Df = 13 Reject H0 Comparing two sample means To find an individual score in a normal distribution, we use a z-score formula: difference between score and mean Z standard deviation To place a sample mean in a distribution of means, we used a t-test formula: difference between 2 sample means t standard error of difference s between means xe xc t obs S xe xc Example for independent-sample ttest Research hypothesis: no effect of group on listening comprehension P=.05, 2-tailed df = (n1-1)+(n2-1)=36 The t critical value for df 30= 2.042 (p<.05), 2.750 (p<.01), Group n mean SD Con. 19 11.7 4.0 Exper. 19 10.5 4.7 Calculation of t-value and conclusion xe xc t obs S xe xc xe xc S xe xc 4.7 2 19 Se2 Sc2 Ne Nc 4.02 19 1.42 1.2 t obs .845 1.42 Conclusion: The 2 groups came from a homogeneous group. Assumptions underlying t-test 1. 2. 3. 4. 5. There are only 2 levels of one independent variable to compare. Each subject is assigned to one and only one group. The data are truly continuous. The mean and SD are the most appropriate measures to describe the data. The distribution in the respective population from which the samples were drawn is normal, and variances are equivalent. Paired-sample t-test df=npairs-1 S D : Standard error of differences between the 2 means n=the number of pairs D=difference between pairs x1 x 2 t SD SD SD n n D D 2 1 n 1 2 n No pretest posttest D D2 1 21 33 12 144 2 17 17 0 0 3 22 30 8 64 4 13 23 10 100 5 33 36 3 9 6 20 25 5 25 7 19 21 2 4 8 14 19 5 25 9 20 19 -1 1 10 31 35 4 16 ∑X=21 0 ∑D2=38 ∑Y=258 ∑D=48 8 Mean= 21 Mean= 258 df=10-1=9 n= 10 t-value=-3.63 t critical value=2.262 Conclusion: H0 rejected 388 (1 10)( 48) 2 SD 4.18 10 1 4.18 SD 1.323 10 X1 X2 21 25.8 t obs 3.63 SD 1.323 Interpretation of results from paired-sample t-test Group n Mean SD t value df p Pretest 38 17.3 5.1 -2.81 37 .008 Post 38 19.4 4.9 1. t critical value for df 30 =2.042 2. Conclusion: H0 rejected One-way ANOVA The one-way ANOVA is used to compare the means of more than 2 groups on one variables in order to see whether the differences are caused by chance or by treatment effect. H0: mean1= mean2 = mean3… F value Error variability = within-group variance = S2within Error variability + treatment effect = between–group variance =S2between If S2between = S2within, then we know our treatments are all similar. If S2between > S2within, we can say there is treatment difference. Fobs= S2between / S2within When F value=1, no effect for treatment, when F value > 1, effect for treatment Example for one-way ANOVA Natural Trans Silent Sit/Fun Drama 1 16 15 14 14 10 2 14 13 13 10 8 3 10 12 15 9 10 4 13 13 17 11 9 5 12 13 11 11 12 6 20 20 11 12 5 7 20 19 12 10 8 8 23 22 10 13 8 9 19 19 13 9 7 10 18 17 12 8 9 ∑X 165 163 128 107 86 ∑X2 2879 2771 1678 1177 772 N=50; ∑X=649; ∑X2=9277; (∑X)2=421201 5 steps for calculation of F value 1. 2. 3. 4. 5. Square each score and add to determine the sum of squares total (SST) Find the sum of squares between groups (SSB) SSB=[(∑X1)2÷ n1 …+(∑Xk)2÷nk] - (∑X)2÷N Find the sum of squares within each group (SSW). SSW=SST-SSB Average each of SSW and SSB to make them sensitive to their respective degrees of freedom. The result is the variance values. S2B=SSB÷(K-1); S2W=SSW÷(N-K) Calculate the F ratio: S2B÷S2W Results of F ration calculation 1. 2. 3. 4. 5. N=50; ∑X=649; ∑X2=9277; (∑X)2=421201 SST= ∑X2-(∑X)2/50 = 9277-421201/50 = 852.98 SSB=(2722.5+2656.9+1634.4+1144.9+739.6) -8424.2 =478.28 SSW= 852.98-478.28=374.7 S2B=SSB÷(K-1)=119.57; S2W=SSW÷(N-K)=8.33 F ratio: S2B÷S2W = 14.35 df in numerator S2B: K-1=4; df in denominator S2W: N-K =45. Check the intersection of 4 and 45 on the F distribution chart: Critical value for 44 df is 2.58 Conclusion: H0 is rejected at the .05 level. SPSS results Source of variance ANOVA score Between Groups Within Groups Total Sum of Squares 478.280 374.700 852.980 df 4 45 49 Mean Square 119.570 8.327 F 14.360 Sig. .000 Assumptions underlying ANOVA Type of variables Independent observations Normal distribution Equal variances of scores in each group A minimum of at least 5 observations per cell. Group differences: multiple comparison 2 ways to precisely locate differences among means: planning the comparison ahead of time (a priori) and post hoc comparison of means (post hoc). In the first case, there are preplanned comparisons and hypotheses (directional) to test. In the second, exploratory comparisons are made and the analyses will be 2-tailed tests.