A one-sample t test can be used to compare a sample mean to a given value. This example, taken from Huntsberger and Billingsley (1989, p. 290), tests whether the mean length of a certain type of court case is 80 days using 20 randomly chosen cases. The data are read by the following DATA step:
title 'One-Sample t Test'; data time; input time @@; datalines; 43 90 84 87 116 95 86 99 93 92 121 71 66 98 79 102 60 112 105 98 ; run;
The only variable in the data set, time , is assumed to be normally distributed. The trailing at signs (@@) indicate that there is more than one observation on a line. The following code invokes PROC TTEST for a one-sample t test:
proc ttest h0=80 alpha=0.1; var time; run;
The VAR statement indicates that the time variable is being studied, while the H0= option specifies that the mean of the time variable should be compared to the value 80 rather than the default null hypothesis of 0. This ALPHA= option requests 10% confidence intervals rather than the default 5% confidence intervals. The output is displayed in Figure 77.1
One-Sample t Test The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Variable N Mean Mean Mean Std Dev Std Dev Std Dev Std Err Minimum Maximum time 20 82.447 89.85 97.253 15.2 19.146 26.237 4.2811 43 121 T-Tests Variable DF t Value Pr > t time 19 2.30 0.0329
Summary statistics appear at the top of the output. The sample size (N), the mean and its confidence bounds (Lower CL Mean and Upper CL Mean), the standard deviation and its confidence bounds (Lower CL Std Dev and Upper CL Std Dev), and the standard error are displayed with the minimum and maximum values of the time variable. The test statistic, the degrees of freedom, and the p -value for the t test are displayed next ; at the 10% ± -level, this test indicates that the mean length of the court cases are significantly different from 80 days ( t =2 . 30 ,p =0 . 0329).
If you want to compare values obtained from two different groups, and if the groups are independent of each other and the data are normally distributed in each group, then a group t test can be used. Examples of such group comparisons include
test scores for two third-grade classes, where one of the classes receives tutoring
fuel efficiency readings of two automobile nameplates, where each nameplate uses the same fuel
sunburn scores for two sunblock lotions, each applied to a different group of people
political attitude scores of males and females
In the following example, the golf scores for males and females in a physical education class are compared. The sample sizes from each population are equal, but this is not required for further analysis. The data are read by the following statements:
title 'Comparing Group Means'; data scores; input Gender $ Score @@; datalines; f 75 f 76 f 80 f 77 f 80 f 77 f 73 m 82 m 80 m 85 m 85 m 78 m 87 m 82 ; run;
The dollar sign ($) following Gender in the INPUT statement indicates that Gender is a character variable. The trailing at signs (@@) enable the procedure to read more than one observation per line.
You can use a group t test to determine if the mean golf score for the men in the class differs significantly from the mean score for the women. If you also suspect that the distributions of the golf scores of males and females have unequal variances, then submitting the following statements invokes PROC TTEST with options to deal with the unequal variance case.
proc ttest cochran ci=equal umpu; class Gender; var Score; run;
The CLASS statement contains the variable that distinguishes the groups being compared, and the VAR statement specifies the response variable to be used in calculations. The COCHRAN option produces p -values for the unequal variance situation using the Cochran and Cox(1950) approximation . Equal tailed and uniformly most powerful unbiased (UMPU) confidence intervals for ƒ are requested by the CI= option. Output from these statements is displayed in Figure 77.2 through Figure 77.4.
Comparing Group Means The TTEST Procedure Statistics UMPU Lower CL Upper CL Lower CL Lower CL Variable Gender N Mean Mean Mean Std Dev Std Dev Std Dev Score f 7 74.504 76.857 79.211 1.6399 1.5634 2.5448 Score m 7 79.804 82.714 85.625 2.028 1.9335 3.1472 Score Diff (1-2) 9.19 5.857 2.524 2.0522 2.0019 2.8619 Statistics UMPU Upper CL Upper CL Variable Gender Std Dev Std Dev Std Err Minimum Maximum Score f 5.2219 5.6039 0.9619 73 80 Score m 6.4579 6.9303 1.1895 78 87 Score Diff (1-2) 4.5727 4.7242 1.5298
T-Tests Variable Method Variances DF t Value Pr > t Score Pooled Equal 12 3.83 0.0024 Score Satterthwaite Unequal 11.5 3.83 0.0026 Score Cochran Unequal 6 3.83 0.0087
Equality of Variances Variable Method Num DF Den DF F Value Pr > F Score Folded F 6 6 1.53 0.6189
Simple statistics for the two populations being compared, as well as for the difference of the means between the populations, are displayed in Figure 77.2. The Variable column denotes the response variable, while the Class column indicates the population corresponding to the statistics in that row. The sample size (N) for each population, the sample means (Mean), and lower and upper confidence bounds for the means (Lower CL Mean and Upper CL Mean) are displayed next. The standard deviations (Std Dev) are displayed as well, with equal tailed confidence bounds in the Lower CL Std Dev and Upper CL Std Dev columns and UMPU confidence bounds in the UMPU Upper CL Std Dev and UMPU Lower CL Std Dev columns . In addition, standard error of the mean and the minimum and maximum data values are displayed.
The test statistics, associated degrees of freedom, and p -values are displayed in Figure 77.3. The Method column denotes which t test is being used for that row, and the Variances column indicates what assumption about variances is being made. The pooled test assumes that the two populations have equal variances and uses degrees of freedom n 1 + n 2 ˆ’ 2, where n 1 and n 2 are the sample sizes for the two populations. The remaining two tests do not assume that the populations have equal variances. The Satterthwaite test uses the Satterthwaite approximation for degrees of freedom, while the Cochran test uses the Cochran and Cox approximation for the p -value.
Examine the output in Figure 77.4 to determine which t test is appropriate. The 'Equality of Variances' test results show that the assumption of equal variances is reasonable for these data (the Folded F statistic F ² = 1 . 53, with p = 0 . 6189). If the assumption of normality is also reasonable, the appropriate test is the usual pooled t test, which shows that the average golf scores for men and women are significantly different ( t = ˆ’ 3 . 83, p = 0 . 0024). If the assumption of equality of variances is not reasonable, then either the Satterthwaite or the Cochran test should be used.
The assumption of normality can be checked using PROC UNIVARIATE; if the assumption of normality is not reasonable, you should analyze the data with the nonparametric Wilcoxon Rank Sum test using PROC NPAR1WAY.