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Chapter 4 Investigating the Difference in Scores ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved Chapter Objectives After completing this chapter, you should be able to 1. Define and give examples of dependent variable, independent variable, null hypothesis, two-tailed test, one-tailed test, degrees of freedom, level of significance, standard error of the mean, and standard error of the difference between means. 2. Define Type 1 and Type II errors. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-2 Chapter Objectives 3. Use and interpret the t-test for independent groups and the t-test for dependent groups. 4. Use and interpret analysis of variance for independent groups and analysis of variance for repeated measures. 5. Use Tukey’s honestly significant difference test (HSD). ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-3 Dependent and Independent Variables Dependent variable • Response to treatment (independent variable) • Variable that is observed for changes as result of treatment Independent variable • Treatment in a study • Controlled by researcher Study to determine if change in blood pressure (dependent variable) after reduction in salt intake (independent variable) ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-4 Testing for Significance Difference Between Two Means Examples: Fitness programs, methods of instruction, flexibility programs Hypothesis: a prediction about the difference between two or more variables Null hypothesis: predicts there will be no statistical difference between the means of groups (H0: X1 = X2) Alternative hypothesis: predicts there will be a difference in means (H1: X1 = X2) ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-5 Testing for Significance Difference Between Two Means Alternative hypothesis used for a two-tailed test; difference in means can be in either direction. One-tailed test used when mean difference can occur only in one direction. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-6 Degrees of Freedom (df) df - concept used in all statistical tests; calculated by subtracting 1 from N (N - 1). Determined by sample size. Indicate the number of scores in a distribution that are free to vary. When using a t-test (two groups), the degrees of freedom equal (N1 - 1) + (N2 - 1) = N1 + N2 - 2 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-7 Level of Significance Probability of rejecting a null hypothesis when it is true. Two most common levels of significance are .01 and .05. .01 = If you reject the null hypothesis, there is 1 chance in 100 that you are rejecting the null hypothesis when it is actually true. .05 = If you reject the null hypothesis, there are 5 chances in 100 that you are in error. Type I error - reject a hypothesis when it is true Type II error - accept the hypothesis when it is not true ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-8 Level of Significance Using the .01 level of significance, rather than the .05 level, reduces the risk of making a Type I error but increases the probability of making a Type II error. Level of significance below .05 rarely uses. Best way of reducing the probability of making a Type II error is to increase the sample size. Level of significance and df are used together to determine the value that a statistical test must yield to reject the null hypothesis. Use values in Appendix B to determine if you reject or accept the null hypothesis. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-9 Standard Error of the Mean (SEM) Distribution of sample means If a large number of equal-size samples were randomly drawn from the same population and formed into a distribution, we would have a sampling distribution of means. If means are in close agreement, value of SEM is small; more confident that any one mean is near the value of the population mean. Not practical to calculate the SEM from a sampling distribution of means; use formula to estimate of SEM. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-10 Standard Error of the Mean (SEM) SEM = s N See Table 4.1 Group running three days/week: X = 80.9 and SEM = 0.38 SEM is added to and subtracted from X (80.9 0.38) If this study were repeated a large number of times, 68.26% of the time the means for the group running three times/week would be in the range of 80.52 to 81.28. For group running five days/week, 68.26% of the time the means would be in the range of 90.51 to 92.09 (91.3 0.79) ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-11 Standard Error of the Mean (SEM) Range interpreted as the limits of the 68% confidence intervals for mean. 95% confidence intervals - add and subtract two SEMs 99% confidence intervals – add and subtract three SEMs ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-12 Standard Error of the Difference Between Means Estimate of the size difference to be expected between two sample means randomly drawn from the same population. To determine standard error of the difference between means, it is necessary to square the SEM of each group, add the results, and find the square root of the sum. Sx-x = SEM12 + SEM22 Table 4.1: Difference between the means is 10.4 and Sx-x = 0.87. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-13 t-Test for Independent Groups Used to determine the significance of the difference between two independent means. Independent means - one group, or variable, does not influence the other group (variable); groups are not related. Assumptions: 1. Initially the two sample groups come from the same population. 2. The population is normally distributed. 3. The two groups are representative samples; that is, they have approximately equal variances. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-14 t-Test for Independent Groups Example: Determine if running 5 days/week would develop cardiorespiratory endurance better than running 3 days/week (see table 4.1). Steps for use of t-test for independent groups Null hypothesis: X1 = X2 Randomly select individuals for each group and running program. 1. Calculate the mean and standard deviation for each group. X1 = 80.9 X2 = 91.3 s1 = 1.20 s2 = 2.50 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-15 t-Test for Independent Groups 2. Calculate the SEM for each group. SEM1 = s1 = 1.2 = 0.38 SEM2 = 0.79 N1 10 3. Calculate the standard error of difference between groups. sx-x = SEM12 + SEM22 = (0.38)2 + (0.79)2 = 0.87 4. Calculate the t-ratio by substituting the values in the formula t = X1 - X2 = 80.9 - 91.3 = -11.95 sx-x 0.87 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-16 t-Test for Independent Groups 5. Determine the degrees of freedom (df). df = 10 + 10 – 2 = 18 6. Refer to t-values in appendix B. If the computed t-ratio is equal to or greater than the critical value in appendix B, reject the null hypothesis. If the ratio is less than the critical value, accept the null hypothesis. t = -11.95 (ignore negative sign) table value (appendix B); 18 df, .05 level = 2.101, .01 level = 2.878 7. Reject the null hypothesis; significant difference in means at .01 level; running 5 days/week develops cardiorespiratory endurance better than running 3 days/week. 4-17 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved t-Test for Dependent Groups Use to determine significant difference between two groups when two groups are not independent but are related to each other. Assumptions: 1. The paired differences are random sample from a normal population. 2. The equal variances assumption is unnecessary, since you will be working with one group. Example: Determine if participation in a basketball class will improve the scores of ninth-grade girls on a speed spot shooting test. (See table 4.2). ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-18 t-Test for Dependent Groups Steps for use of t-test for dependent groups Null hypothesis: There will be no difference in means of speed spot shooting pretest and posttest (X1 = X2). Administer test first day of class and again at the conclusion of the basketball unit. 1. List the pairs of scores so that you can subtract one from the other. 2. Label a column D and determine the difference for each pair of scores. 3. Label a column D2, square each D, and sum D2 (D2). D2 = 50 4. Calculate the mean difference (D). D = 18; D = 1.8 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-19 t-Test for Dependent Groups 5. Calculate the standard deviation of the difference. sD = D2 - N(D2) = 50 - 10(1.82) = 1.33 N 10 6. Calculate the standard error of the difference (sD). The sD is equivalent to the SEM and is obtained in a similar manner. Formula is: sD = sD = 1.33 = .42 N 10 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-20 t-Test for Dependent Groups 7. Calculate the t-ratio by substituting the values in the formula t = D = 1.8 = 4.29 sD .42 8. Determine the degrees of freedom. df = 10 - 1 = 9. 9. Refer to the t-values in appendix B. With 9 df, the t-ratio of 4.29 is greater than the t-value of 2.262 needed for significance at the .05 level of significance and greater than the t-value of 3.250 needed at 01 level. 10. Reject the null hypothesis at the .01 level of significance. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-21 Testing for Significant Difference Among Three or More Means Analysis of variance (ANOVA) is used to test for significant difference among three or more means. Special Terms and Symbols N = number of scores n = the number of scores in a group k = the number of independent groups or the number of trials (measures) performed on the same subjects (repeated measures) ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-22 ANOVA – Special Terms and Symbols r = number of rows or subjects grand X = sum of all the scores in all groups grand X2 = sum of the square of all scores in all groups total sum of squares (SST) = the sum of the squared deviations of every score from the grand X; represents the variability of each score in all groups from the grand X sum of squares within groups (SSW) = the sum of the squared deviations of each score from its group X; also referred to as sum of squares for error ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-23 ANOVA – Special Terms and Symbols sum of squares between groups (SSB) = the sum of the squared deviations of each group X from the grand X; also referred to as treatment sum of squares mean square for the sum of squares within groups (MSW) = variance within the groups mean square for the sum of squares between groups (MSB) = variance between groups F distribution = table values required to reject null hypothesis ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-24 ANOVA – Special Terms and Symbols degrees of freedom within groups = (N - k); total number of measures or scores (N) minus number of groups (k); degrees of freedom for denominator in F distribution degrees of freedom between groups = (k - 1); number of groups (k) minus 1; degrees of freedom for numerator in F distribution degrees of freedom between groups = k - 1 degrees of freedom within groups = N - k ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-25 ANOVA for Independent Groups Used when the sample groups are not related to each other. Assumptions 1. The samples are randomly drawn from a normally distributed population. 2. The variances of the samples are approximately equal. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-26 ANOVA for Independent Groups Example: Determine if there is a difference in three programs designed to improve trunk extension. (See table 4.3). Null hypothesis: X1 = X2 = X3 Instructor randomly assigns subjects to one of three groups; each group participates in a different trunk extension program; all subjects are given the same trunk extension test. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-27 Steps for Calculation of ANOVA for Independent Groups 1. Sum each group of scores to get X1, X2, and X3. Add the group sums to get a grand X (X1 + X2 + X3 = grand X). 171 + 154 + 152 = 2904 2. Square each score and sum the squared scores of each group (X2). Add the X2 for each group to get a grand X2 (X12 + X22 + X32 = grand X2). 3665 + 2972 + 2904 = 9541 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-28 Steps for Calculation of ANOVA for Independent Groups 3. Calculate the correction factor (C). C is necessary because the raw scores are used rather than deviations of the scores from the mean. C = (grand X)2 C = (477)2 = 9480.38 total N 24 4. Calculate the total sum of squares (SST). SST = grand X2 - C SST = 9541 - 9480.38 = 60.62 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-29 Steps for Calculation of ANOVA for Independent Groups 5. Calculate the sum of squares between groups (SSB). SSB = (X1)2 + (X2)2 + (X3)2 - C n1 n2 n3 SSB = (171)2 + (154)2 + (152)2 - 9480.38 = 27.25 8 8 8 6. Calculate the sum of squares within groups (SSW) SSW = SST - SSB SSW = 60.62 - 27.25 = 33.37 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-30 Steps for Calculation of ANOVA for Independent Groups 7. Calculate the mean square for the sum of squares between groups. MSB = SSB k-1 MSB = 27.25 = 13.63 3–1 8. Calculate the mean square for the sum of squares within groups. MSW = SSW N-k MSW = 33.37 = 1.59 24 - 3 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-31 Steps for Calculation of ANOVA for Independent Groups 9. Calculate the F-ratio (see table 4.4). F = MSB MSW F = 13.63 = 8.57 1.59 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-32 Steps for Calculation of ANOVA for Independent Groups 10. Refer to the F-distribution in appendix C to determine if F is significant. Numerator degrees of freedom = (k - 1) = ((3 - 1) = 2 Denominator degrees of freedom = (N - k) = (24 - 3) = 21 .05 table value = 3.47 .01 table value = 5.78 F-ratio of 8.57 greater than 5.78; significant difference in three means at the .01 level. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-33 Post Hoc Test If significant difference in means, must determine which means are significantly different from each other. Test used for this purpose is called post hoc test. Tukey’s honestly significant difference test (HSD) HSD = q(, k, N-k) MSW (used if n’s in each group are n equal) HSD = q(, k, N-k) MSW 1 + 1 (used if n’s in any two 2 2 n 1 n2 groups are unequal) ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-34 Post Hoc Test Since n’s are equal in table 4.3, use first formula. Use appendix D to fine q. q = about 4.64 (k = 3, N - k = 21); since table jumps from 20 to 24 for the denominator, will use 3,20. HSD = 4.64 1.59 = 4.64 0.19875 = 2.069 8 2.069 represents the minimum raw-score difference between any two means that may be declared significant. X1 - X2 = 21.38 - 19.25 = 2.13* X2 - X3 = 19.25 - 19.00 = 0.25 X1 - X3 = 21-38 - 19.00 = 2.38* (*significant difference) ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-35 Analysis of Variance for Repeated Measures Used when repeated measures are made on the same subjects. Basic assumptions: 1. The samples are randomly selected from a normal population. 2. The variances for each measurement are approximately equal. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-36 Analysis of Variance for Repeated Measures Example: Determine if loudness of sound is a factor in the making of foul shots (see table 4.5). Null hypothesis: X1 = X2 = X3 In this technique, N refers to the total number of measurements. 10 subjects are measured 3 times each, so N = 30. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-37 Steps in ANOVA for Repeated Measures 1. Calculate the grand X and grand X2. X1 + X2 + X3 = grand X 100 + 110 + 94 = 304 X12 + X22 + X32 = grand X2 1016 + 1224 + 890 = 3130 2. Calculate the correction factor. C = (grand X)2 = (304)2 = 3080.53 N 30 3. Calculate the total sum of squares (SST). SST = grand X2 - C SST = 3130 - 3080.53 = 49.47 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-38 Steps in ANOVA for Repeated Measures 4. Calculate the sum of squares between groups (SSB); this is the sum of squares between trials. SSB = (X1)2 + (X2)2 + (X3)2 - C n1 n2 n3 SSB = (100)2 + (110)2 + (94)2 - 3080.53 = 13.07 10 10 10 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-39 Steps in ANOVA for Repeated Measures 5. Calculate the sum of squares to determine the effects of the test-retest (SSsubjects). Add the scores of the three trials for each subject, square the sum, and add the squared sums for each subject. The sum of the squared sums for each subject is divided by k (the number of trials), and C is subtracted from the resulting value. SSsubjects = ( rows)2 – C k = (11+12 +10)2 + (9+10+9)2 + . . . (9 +10 +10)2 - 3080.53 3 SSsubjects = 3108.67 – 3080.53 = 28.14 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-40 Steps in ANOVA for Repeated Measures 6. Compute the sum of squares for error (also referred to as within-group sum of squares). This calculation represents the sum of the squared deviations of each score from its group X. SSE = SST - SSB - Sssubjects SSE = 49.47 - 13.07 - 28.14 = 8.26 7. Calculate the mean square for the sum of squares between groups (trials). MSB = SSB = 13.07 = 6.54 k–1 3-1 ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-41 Steps in ANOVA for Repeated Measures 8. Calculate the mean square for the sum of squares between subjects. MSsubjects = Sssubjects r-1 MSsubjects = 28.14 = 3.13 10 - 1 9. Calculate the mean square for the sum of squares for error. MSE = SSE (k - 1)(r - 1) MSE = 8.26 = 0.46 (2)(9) ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-42 Steps in ANOVA for Repeated Measures 10. Calculate the F-ratio (see table 4.6). F-ratio = MSB MSE F-ratio = 6.54 = 14.22 0.46 11. Refer to the F-distribution in appendix C. The degrees of freedom for the numerator are (k-1) = (3-1) = 2. For the denominator, the degrees of freedom are (k-1)(r-1) = (3-1)(10-1) = 18. The F-ratio of 14.22 is greater than the 6.01 value needed for significance at .01 level, so we reject the null hypothesis. Now use Tukey’s HSD test to determine which means are significantly different. ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-43 Post Hoc Test Tukey’s HSD test for ANOVA for repeated measures HSD = q(, k, [k-1][r-1]) MSE r q = value obtained from table in appendix D = 4.70 = significance level = .01 k = number of trials or measures performed on the same subjects = 3 r = number of rows or number of subjects = 10 (k-1)(r-1) = degrees of freedom for denominator in appendix D = (3-1)(10-1) = 18 MSE = mean square for the sum of squares for error = 0.46 (table 3.8) ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-44 Post Hoc Test HSD = q(, k, [k-1][r-1]) MSE r HSD = 4.70 0.46 = 1.008 10 X1 - X2 = 10 - 11 = -1 X2 - X3 = 11- 9.4 = 1.6* X1 - X3 = 10 - 9.4 = 0.6 *significant difference between X2 (medium sound) and X2 (loud sound) ©2013, The McGraw-Hill Companies, Inc. All Rights Reserved 4-45