Handout 1: One-sample, paired, and two-sample t-tests

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1 Stat120C Handout 1: One-sample, paired, and two-sample t-tests Instructor: Zhaoxia Yu In this handout we show how to use R to conduct t-tests. 1 One sample t-test Consider the example we discussed in class: we have a sample from a normal distribution and the observed values are: 22, 25, 36, 32, 49, 46, 31, 52, 30, 42, 44 Suppose we want to test H 0 : µ = 30 v.s. H 1 : µ 30. To conduct a one-sample t-test, we first compute the sample mean and variance: X = 1 n The statistic of the one-sample t-test is xi = 33.73, s 2 = ˆσ 2 = 1 (xi n 1 X) 2 = t = X µ 0 s2 /n = /11 = 1.10 From Appendix B (TABLE 4) of our text book, the upper 97.5th percentile of the t distribution with 10 degrees of freedom is Because t < 2.23, we do not reject the null hypothesis. Below is the computation in R: #enter data x = c(19, 26, 32, 24, 49, 42, 23, 53, 26, 39, 38) #get the sample size n=length(x) #calculate the sample mean and variance mean(x) [1]

2 2 Handout 1: One-sample, paired, and two-sample t-tests var(x) [1] #compute the t statistic t.stat= (mean(x)-30)/sqrt(var(x)/n) # print the statistic t.stat [1] #find the 97.5th percentile of the t-distribution with 10 degrees of freedom qt(0.975, 10) [1] #alternatively, you can compute the two-sided p-value 2*(1-pt(abs(t.stat),10)) [1] R has a built-in function t.test. Here shows how to use t.test to conduct one-sample t-test in R t.test(x, mu=30) One Sample t-test data: x t = , df = 10, p-value = alternative hypothesis: true mean is not equal to mean of x In the above R output, you have the sample mean, the 95% C.I., the t-test statistic and the p-value. You can use help(t.test) in R to see options of the function. For example, you can use t.test(x, mu=30, alternative= greater ) to test H 0 : µ = 30 v.s. H 1 : µ 30.

3 Handout 1: One-sample, paired, and two-sample t-tests 3 2 Paired t-test Consider the before and after measurements problem and suppose the following data are observed: before after We want to test H 0 : µ before = µ after v.s. H 0 : µ before µ after. We provide three methods of conducting a paired t-test #enter data weight.before = c(19, 26, 32, 24, 49, 42, 23, 53, 26, 39, 38) weight.after = c(22, 25, 36, 32, 49, 46, 31, 52, 30, 42, 44) # define a new variable "diff" diff = weight.after - weight.before n = length(diff) #method 1: compute the t-test statistic then find the critical value mean.diff=mean(diff) var.diff =var(diff) t.stat.diff=mean.diff/sqrt(var.diff/n) t.stat.diff [1] #the critical value qt(0.975, 10) [1] #the p-value 2*(1-pt(abs(t.stat.diff), 10)) [1]

4 4 Handout 1: One-sample, paired, and two-sample t-tests #method 2: use the R built-in function t.test(diff, mu=0) One Sample t-test data: diff t = , df = 10, p-value = alternative hypothesis: true mean is not equal to mean of x #method 3, use the R built-in function in another way t.test(weight.after, weight.before, paired=t) Paired t-test data: weight.after and weight.before t = , df = 10, p-value = alternative hypothesis: true difference in means is not equal to mean of the differences Two-sample t-test The pair t-test we just showed is for paired data. Suppose you forget about the fact that the data are paired and your treat the two samples as two independent samples. 3.1 Two-sample t-test with equal variance #the difference of sample means mean(weight.before) - mean(weight.after) [1]

5 Handout 1: One-sample, paired, and two-sample t-tests 5 #the standard deviation of the difference in sample means sp = sqrt( ((n-1)*var(weight.before) + (n-1)*var(weight.after)) / (n + n -2)) #the test statistic and p-value t.stat = (mean(weight.after) - mean(weight.before)) / (sp*sqrt(1/n+1/n)) p.value.unpaired = 2*(1-pt(abs(t.stat), df=(n+n-2))) p.value.unpaired [1] #you can also use the R built-in function t.test to conduct a two-sample t-test #with the assumption that the two populations have the same variance t.test(weight.after, weight.before, var.equal=t) Two Sample t-test data: weight.after and weight.before t = , df = 20, p-value = alternative hypothesis: true difference in means is not equal to mean of x mean of y Two-sample t-test without the equal-variance assumption - Welch two-sample t-test In practice, it is not always reasonable to assume equal variance. Below shows how to do a two-sample t-test without the equal-variance assumption: t.test(weight.after, weight.before) Welch Two Sample t-test data: weight.after and weight.before t = , df = , p-value = alternative hypothesis: true difference in means is not equal to mean of x mean of y

6 6 Handout 1: One-sample, paired, and two-sample t-tests In the previous section we saw that the paired t-test rejects the null hypothesis at level 0.05; however, the two-sample t-tests in this section give different answers! As discussed in page 445 of Rice, when samples are paired and there is a positive correlation, ignoring the pairing may lead to inefficient use of data. Type help(t.test) on the R console to see more options. The R source code can be found here R source code