CHAPTER 7 PARAMETRIC HYPOTHESIS TESTS A wide variety of the commonly used parametric hypothesis tests are available for use right on screen. They provide quick and easy solutions to the computation of various parametric tests that are often very difficult to perform by hand or by use of a calculator. The computations are rapid and very accurate. All you need to do is enter a few items of information and you will quickly have the answers you need. None of the parametric hypothesis tests will send the results to your printer. However, all of your results are saved for you as you work along. Then, when you exit the SPPC you will be given the option of reviewing your work on screen, printing it, or storing it in a disk file of your choice. ONE-SAMPLE TEST OF MEANS When you choose the option to conduct a one-sample test of means, you will be conducting a t-test of the hypothesis that a sample mean significantly differs from 0 or from a known or presumed population mean. The program will ask you to enter the sample size, the sample mean, the standard deviation computed from the sample and corrected for bias, and the population mean. If you enter a population mean of 0 you will be testing the hypothesis that the sample mean departs from zero. If you enter any other population mean, Mu, you will be testing the hypothesis that the Diff = Mn - Mu = 0. You may enter raw or summary data at the keyboard and you may conduct your hypothesis tests at the 90%, 95% or the 99% confidence level. The following are sample outputs from the one-sample test of means where Mean is the sample mean, Mu is the population mean, and SEM is the standard error of the mean. N = 7 SEM = 5.37568 Mean = 27.57143 t = 5.12892 Mu = 0.00000 p <= 0.00268 99% Confidence Interval 8.76192 <= Mn <= 46.38094 N = 189 SEM = 0.80013 Mean = 35.00000 t = 43.74277 Mu = 0.00000 p <= 0.00000 99% Confidence Interval 32.93886 <= Mn <= 37.06114 N = 300 SEM = 1.21244 Mean = 47.00000 t = -0.82479 Mu = 48.00000 p <= 0.20475 95% Confidence Interval 44.62363 <= Diff <= 49.37637 TWO-SAMPLE TEST OF MEANS: DEPENDENT SAMPLES The two-sample test of means for dependent samples enables you to compare two means that are obtained from the same subjects. It is often called the "direct difference t-test of means". If you enter raw scores, the program will compute and report the correlation for you. If you enter summary data, you must know the correlation between the two variables as well as the sample means, standard deviations and the sample size. The program produces the difference between the two means, the standard error of that difference, the t-test and its associated probability, and one of three confidence intervals that you specify. The following are samples of outputs for the two-sample test of means using dependent samples. GROUP I GROUP II N = 34 N = 34 Mean x = 87.00000 Mean x = 95.00000 SDx = 23.00000 SDx = 28.00000 SEDiff = 5.07323 r = 0.34000 t = -1.57691 p <= 0.12070 Mean 1 - Mean 2 = -8.00000 99% Confidence Interval -21.95138 <= Diff <= 5.95138 TWO-SAMPLE TEST OF MEANS: INDEPENDENT SAMPLES The two-sample test of means using independent samples is the good old "t-test for mean differences". You may enter raw or summary data from the keyboard and the following are sample outputs from this procedure. GROUP I GROUP II N = 89 N = 114 Mean x = 54.00000 Mean x = 61.00000 SDx = 28.00000 SDx = 31.00000 SEDiff = 4.20442 t = -1.66492 p <= 0.04796 Mean 1 - Mean 2 = -7.00000 95% Confidence Interval -15.24066 <= Diff <= 1.24066 TEST PEARSON CORRELATIONS There are six tests of correlations that have identical inputs and outputs. These are the test of a Pearson r, Phi, Point-Biserial, Spearman's Rho, partial correlations, and semi-partial correlations. For each of these tests, you need only enter the sample size and the value of the correlation. The program does the rest and provides you with an F-ratio, degrees of freedom for hypothesis (dfh), degrees of freedom for error (dfe), the associated probability for the F-ratio and a confidence interval for your correlation. N = 18 dfh = 1 r = 0.46000 dfe = 16 F = 4.29427 p <= 0.05229 95% Confidence Interval -0.05003 <= r <= 0.77973 TEST PHI COEFFICIENT See the above explanation for "Test Pearson Correlation". The inputs and outputs are identical. The only difference is that you will enter your Phi coefficient instead of the Pearson r. N = 67 dfh = 1 r = 0.18000 dfe = 65 F = 2.17652 p <= 0.14120 95% Confidence Interval -0.06791 <= r <= 0.40698 TEST POINT-BISERIAL CORRELATION See the above explanation for "Test Pearson Correlation". The inputs and outputs are identical. The only difference is that you will enter your Point-Biserial correlation instead of the Pearson r. N = 219 dfh = 1 r = 0.17000 dfe = 217 F = 6.45793 p <= 0.00552 95% Confidence Interval 0.03829 <= r <= 0.29591 TEST SPEARMAN RHO See the above explanation for "Test Pearson Correlation". The inputs and outputs are identical. The only difference is that you will enter your Spearman Rho instead of the Pearson r. N = 189 dfh = 1 r = 0.67000 dfe = 187 F = 152.32136 p <= 0.00000 95% Confidence Interval 0.58302 <= r <= 0.74179 TEST PARTIAL CORRELATION See the above explanation for "Test Pearson Correlation". The inputs and outputs are identical. The only difference is that you will enter your partial correlation instead of the Pearson r. N = 118 dfh = 1 r = 0.23000 dfe = 116 F = 6.47915 p <= 0.01178 95% Confidence Interval 0.04951 <= r <= 0.39594 TEST SEMI-PARTIAL CORRELATION See the above explanation for "Test Pearson Correlation". The inputs and outputs are identical. The only difference is that you will enter your semi-partial correlation instead of the Pearson r. N = 56 dfh = 1 r = 0.54000 dfe = 54 F = 22.22812 p <= 0.00009 95% Confidence Interval 0.31801 <= r <= 0.70586 TEST MULTIPLE CORRELATIONS The test of a multiple correlation is used to test the null hypothesis, H: R = 0. In order to use it, you must enter the sample size, the multiple correlation, R, and the number of independent variables, IVs, used to obtain R. The program will produce the F-ratio, degrees of freedom for hypothesis (dfh), degrees of freedom for error (dfe), the probability associated with the F-ratio, and the confidence interval that you specifiy. The program then produces the same information for the shrunken multiple correlation, and the following are sample outputs from the program. N = 137 dfh = 3 r = 0.43000 dfe = 133 F = 10.05672 p <= 0.00004 IV's = 3 95% Confidence Interval 0.27989 <= r <= 0.55959 SHRUNKEN CORRELATION N = 137 dfh = 3 r = 0.40806 dfe = 133 F = 10.05672 p <= 0.00004 IV's = 3 95% Confidence Interval 0.25518 <= r <= 0.54104 TEST CHANGE IN R^2 The ability to test a change in a squared multiple correlation, R^2, can be extremely important. In order to do that, you will need to enter the degrees of freedom for hypothesis, dfh, the degrees of freedom for error, dfe, the change in R^2 that you wish to test, and the overall R^2 for the multiple regression model. In the first example shown below, the value of dfh = 1 for the test of an increment in R^2 due to the addition of only one variable. In the second example, the change in R^2 was based on the addition of three variables so the value of dfh = 3. dfh = 1 dfe = 218 R^2 Change = 0.14000 R^2 Total = 0.47000 F-ratio = 57.5849 p <= 0.48394 dfh = 3 dfe = 321 R^2 Change = 0.26000 R^2 Total = 0.37000 F-ratio = 44.1587 p <= 0.00006 ONE-SAMPLE TEST OF PROPORTIONS The one-sample test of proportions is much like the one-sample test of means. If you enter a population proportion of 0 you will be testing the hypothesis that your sample proportion is equal to 0. If you enter any other value for the population proportion, you will be testing the hypothesis that Diff = SP-PP = 0. These features are illustrated by the following sample outputs. SP = 0.14000 PP = 0.00000 N = 89 SEP = 0.03678 t = 2.00000 p <= 0.00051 95% Confidence Interval for Proportion 0.00000 <= SP <= 0.28000 SP = 0.23000 PP = 0.27000 N = 121 SEP = 0.03826 t = 1.98000 p <= 0.29819 95% Confidence Interval for Proportion 0.19000 <= Diff <= 0.27000 TWO-SAMPLE TEST OF PROPORTIONS: DEPENDENT SAMPLES The two-sample test of proportions for dependent samples enables you to test the hypothesis, Ho: p1 = p2. You need only enter the two proportions and the sample size in order to obtain the output illustrated below. GROUP I GROUP II Prop = 0.42000 Prop = 0.38000 N = 134 N = 134 Difference = 0.04000 t = -0.51769 p <= 0.30234 TWO-SAMPLE TEST OF PROPORTIONS: INDEPENDENT SAMPLES The two-sample test of proportions for independent samples enables you to test the hypothesis, Ho: p1 = p2. You need only enter the two proportions and the sample size in order to obtain the output illustrated below. GROUP I GROUP II Prop = 0.38000 Prop = 0.31000 N = 89 N = 112 Std Dev = 0.05025 Std Dev = 0.04479 Difference = 0.07000 SEDiff = 0.06732 t = 1.03989 p <= 0.14920 95% Confidence Interval for Difference -0.06194 <= Diff <= 0.20194 ONE-SAMPLE TEST OF A STANDARD DEVIATION This test is analagous to the one-sample test of means except that it is conducted for standard deviations. N = 67 Mean = 35 SD = 17.00000 PSD = 21.00000 t = -2.72373 SESD = 1.46858 p <= 0.00816 95% Confidence Interval -6.93715 <= Diff <= -1.06285 TWO-SAMPLE TEST OF VARIANCES: DEPENDENT SAMPLES This test is analagous to the two-sample test of means using dependent samples except that the test is applied to variances. GROUP I GROUP II N = 36 N = 36 Var X = 127.00000 Var X = 138.00000 SD X = 11.26943 SD X = 11.74734 Difference = Var1 - Var2 = -11.00000 SEDiff = 43.72152 t = -0.25159 r = 0.27000 p <= 0.79825 95% Confidence Interval -100.27934 <= Diff <= 78.27934 TWO-SAMPLE TEST OF VARIANCES: INDEPENDENT SAMPLES This test is analagous to the two-sample test of means using independent samples except that the test is applied to variances. GROUP I GROUP II N = 168 N = 236 Var x = 489.00000 Var x = 512.00000 F = 1.04703 p <= 0.18867 TEST TWO CORRELATIONS: DEPENDENT SAMPLES Often one has a correlation between X and Y and for X and Z from the same sample. In such cases it is often desirable to test the hypothesis that the two correlations are equal, i.e. that r(xy) = r(xz). This procedure enables you to do that. You need only enter the sample size and the two correlations. However, you must also enter the correlation, r(yz), between the variables Y and Z. Remember, the test is on the hypothesis, Ho: r(xy) = r(xz). The following is a sample of the output from this procedure. N = 79 r(xy) = 0.47000 r(xz) = 0.52000 r(yz) = 0.23000 Diff = r(xy) - r(xz) = -0.05000 z = -0.44074 p <= 0.66481 95% Confidence Interval -2.46105 <= Diff <= 1.57223 TEST TWO CORRELATIONS: INDEPENDENT SAMPLES Often one has two correlations for the same variables but from independent samples. In such cases it is often desirable to test the hypothesis that the two correlations are equal. This procedure enables you to do that. You need only enter the sample size and the correlation for each of the two groups or samples and the following is a sample of the output from this procedure. GROUP I GROUP II N = 89 N = 131 r = 0.46000 r = 0.37000 Z = 0.49731 Z = 0.38842 Difference = 0.10889 t = 0.78096 p <= 0.21741 95% Confidence Interval -0.16439 <= Diff <= 0.38217 OMNIBUS R TEST The Omnibus R Test enables you to enter an entire correlation matrix from the keyboard and then test the hypothesis that there are no significant correlations within the entire matrix. In other words, one is never justified in analyzing a correlation matrix using, multivariate procedures for example, unless one can first reject the null hypothesis, Ho: R = 0. The Omnibus R Test provides such a test and the following is a sample of the output from this procedure for 4x4 correlation matrix. N = 367 df = 6 Chi-Square = 571.82834 p <= 0.00000