Understanding Independent Samples tTest
Statistical Technique in Review
The independent samples ttest is a parametric statistical technique used to determine significant differences between the scores obtained from two samples or groups. Since the ttest is considered fairly easy to calculate, researchers often use it in determining differences between two groups. The ttest examines the differences between the means of the two groups in a study and adjusts that difference for the variability (computed by the standard error) among the data. When interpreting the results of ttests, the larger the calculated t ratio, in absolute value, the greater the difference between the two groups. The significance of a t ratio can be determined by comparison with the critical values in a statistical table for the t distribution using the degrees of freedom (df) for the study (see Appendix A Critical Values for Student’s t Distribution at the back of this text). The formula for df for an independent ttest is as follows:
df=(numberofsubjectsinsample1+numberofsubjectsinsample2)−2
Exampledf=(65insample1+67insample2)−2=132−2=130
The ttest should be conducted only once to examine differences between two groups in a study, because conducting multiple ttests on study data can result in an inflated Type 1 error rate. A Type I error occurs when the researcher rejects the null hypothesis when it is in actuality true. Researchers need to consider other statistical analysis options for their study data rather than conducting multiple ttests. However, if multiple ttests are conducted, researchers can perform a Bonferroni procedure or more conservative post hoc tests like Tukey’s honestly significant difference (HSD), StudentNewmanKeuls, or Scheffé test to reduce the risk of a Type I error. Only the Bonferroni procedure is covered in this text; details about the other, more stringent post hoc tests can be found in Plichta and Kelvin (2013) and Zar (2010).
The Bonferroni procedure is a simple calculation in which the alpha is divided by the number of ttests conducted on different aspects of the study data. The resulting number is used as the alpha or level of significance for each of the ttests conducted. The Bonferroni procedure formula is as follows: alpha (α) ÷ number of ttests performed on study data = more stringent study α to determine the significance of study results. For example, if a study’s α was set at 0.05 and the researcher planned on conducting five ttests on the study data, the α would be divided by the five ttests (0.05 ÷ 5 = 0.01), with a resulting α of 0.01 to be used to determine significant differences in the study.
The ttest for independent samples or groups includes the following assumptions:
1. The raw scores in the population are normally distributed.
2. The dependent variable(s) is(are) measured at the interval or ratio levels.
3. The two groups examined for differences have equal variance, which is best achieved by a random sample and random assignment to groups.
4. All scores or observations collected within each group are independent or not related to other study scores or observations.
The ttest is robust, meaning the results are reliable even if one of the assumptions has been violated. However, the ttest is not robust regarding betweensamples or withinsamples independence assumptions or with respect to extreme violation of the assumption of normality. Groups do not need to be of equal sizes but rather of equal variance. Groups are independent if the two sets of data were not taken from the same subjects and if the scores are not related (Grove, Burns, & Gray, 2013; Plichta & Kelvin, 2013). This exercise focuses on interpreting and critically appraising the ttests results presented in research reports. Exercise 31 provides a stepbystep process for calculating the independent samples ttest.
Research Article
Source
Canbulat, N., Ayhan, F., & Inal, S. (2015). Effectiveness of external cold and vibration for procedural pain relief during peripheral intravenous cannulation in pediatric patients. Pain Management Nursing, 16(1), 33–39.
Introduction
Canbulat and colleagues (2015, p. 33) conducted an experimental study to determine the “effects of external cold and vibration stimulation via Buzzy on the pain and anxiety levels of children during peripheral intravenous (IV) cannulation.” Buzzy is an 8 × 5 × 2.5 cm batteryoperated device for delivering external cold and vibration, which resembles a bee in shape and coloring and has a smiling face. A total of 176 children between the ages of 7 and 12 years who had never had an IV insertion before were recruited and randomly assigned into the equally sized intervention and control groups. During IV insertion, “the control group received no treatment. The intervention group received external cold and vibration stimulation via Buzzy . . . Buzzy was administered about 5 cm above the application area just before the procedure, and the vibration continued until the end of the procedure” (Canbulat et al., 2015, p. 36). Canbulat et al. (2015, pp. 37–38) concluded that “the application of external cold and vibration stimulation were effective in relieving pain and anxiety in children during peripheral IV” insertion and were “quickacting and effective nonpharmacological measures for pain reduction.” The researchers concluded that the Buzzy intervention is inexpensive and can be easily implemented in clinical practice with a pediatric population.
Relevant Study Results
The level of significance for this study was set at α = 0.05. “There were no differences between the two groups in terms of age, sex [gender], BMI, and preprocedural anxiety according to the self, the parents’, and the observer’s reports (p > 0.05) (Table 1). When the pain and anxiety levels were compared with an independent samples t test, . . . the children in the external cold and vibration stimulation [intervention] group had significantly lower pain levels than the control group according to their selfreports (both WBFC [Wong Baker Faces Scale] and VAS [visual analog scale] scores; p < 0.001) (Table 2). The external cold and vibration stimulation group had significantly lower fear and anxiety 163levels than the control group, according to parents’ and the observer’s reports (p < 0.001) (Table 3)” (Canbulat et al., 2015, p. 36).
TABLE 1
COMPARISON OF GROUPS IN TERMS OF VARIABLES THAT MAY AFFECT PROCEDURAL PAIN AND ANXIETY LEVELS
Characteristic  Buzzy (n = 88)  Control (n = 88)  χ^{2} p 
Sex  
Female (%), n  11 (12.5)  13 (14.8)  .82 
Male (%), n  77 (87.5)  75 (85.2)  .41 
Characteristic  Buzzy (n = 88)  Control (n = 88)  t p 
Age (mean ± SD)  8.25 ± 1.51  8.61 ± 1.69  −1.498 .136 
BMI (mean ± SD)  25.41 ± 6.74  26.94 ± 8.68  −1.309 .192 
Preprocedural anxiety  
Selfreport (mean ± SD)  2.03 ± 1.29  2.11 ± 1.58  −0.364 .716 
Parent report (mean ± SD)  2.11 ± 1.20  2.17 ± 1.42  −0.285 .776 
Observer report (mean ± SD)  2.18 ± 1.17  2.24 ± 1.37  −0.295 .768 
BMI, body mass index.
Canbulat, N., Ayban, F., & Inal, S. (2015). Effectiveness of external cold and vibration for procedural pain relief during peripheral intravenous cannulation in pediatric patients. Pain Management Nursing, 16(1), p. 36.
TABLE 2
COMPARISON OF GROUPS’ PROCEDURAL PAIN LEVELS DURING PERIPHERAL IV CANNULATION
Buzzy (n = 88)  Control (n = 88)  t p 

Procedural selfreported pain with WBFS (mean ± SD)  2.75 ± 2.68  5.70 ± 3.31  −6.498 0.000 
Procedural selfreported pain with VAS (mean ± SD)  1.66 ± 1.95  4.09 ± 3.21  −6.065 0.000 
IV, intravenous; WBFS, WongBaker Faces Scale; SD, standard deviation; VAS, visual analog scale.
Canbulat, N., Ayban, F., & Inal, S. (2015). Effectiveness of external cold and vibration for procedural pain relief during peripheral intravenous cannulation in pediatric patients. Pain Management Nursing, 16(1), p. 37.
TABLE 3
COMPARISON OF GROUPS’ PROCEDURAL ANXIETY LEVELS DURING PERIPHERAL IV CANNULATION
Procedural Child Anxiety  Buzzy (n = 88)  Control (n = 88)  t p 
Parent reported (mean ± SD)  0.94 ± 1.06  2.09 ± 1.39  −6.135 0.000 
Observer reported (mean ± SD)  0.92 ± 1.03  2.14 ± 1.34  −6.745 0.000 
SD, standard deviation; IV, intravenous.
Canbulat, N., Ayban, F., & Inal, S. (2015). Effectiveness of external cold and vibration for procedural pain relief during peripheral intravenous cannulation in pediatric patients. Pain Management Nursing, 16(1), p. 37.
Study Questions
1. What type of statistical test was conducted by Canbulat et al. (2015) to examine group differences in the dependent variables of procedural pain and anxiety levels in this study? What two groups were analyzed for differences?
2. What did Canbulat et al. (2015) set the level of significance, or alpha (α), at for this study?
3. What are the t and p (probability) values for procedural selfreported pain measured with a visual analog scale (VAS)? What do these results mean?
4. What is the null hypothesis for observerreported procedural anxiety for the two groups? Was this null hypothesis accepted or rejected in this study? Provide a rationale for your answer.
5. What is the ttest result for BMI? Is this result statistically significant? Provide a rationale for your answer. What does this result mean for the study?
6. What causes an increased risk for Type I errors when ttests are conducted in a study? How might researchers reduce the increased risk for a Type I error in a study?
7. Assuming that the ttests presented in Table 2 and Table 3 are all the ttests performed by Canbulat et al. (2015) to analyze the dependent variables’ data, calculate a Bonferroni procedure for this study.
8. Would the ttest for observerreported procedural anxiety be significant based on the more stringent α calculated using the Bonferroni procedure in question 7? Provide a rationale for your answer.
9. The results in Table 1 indicate that the Buzzy intervention group and the control group were not significantly different for gender, age, body mass index (BMI), or preprocedural anxiety (as measured by selfreport, parent report, or observer report). What do these results indicate about the equivalence of the intervention and control groups at the beginning of the study? Why are these results important?
10. Canbulat et al. (2015) conducted the χ^{2} test to analyze the difference in sex or gender between the Buzzy intervention group and the control group. Would an independent samples ttest be appropriate to analyze the gender data in this study (review algorithm in Exercise 12)? Provide a rationale for your answer.
Answers to Study Questions
1. An independent samples ttest was conducted to examine group differences in the dependent variables in this study. The two groups analyzed for differences were the Buzzy experimental or intervention group and the control group.
2. The level of significance or alpha (α) was set at 0.05.
3. The result was t = −6.065, p = 0.000 for procedural selfreported pain with the VAS (see Table 2). The t value is statistically significant as indicated by the p = 0.000, which is less than α = 0.05 set for this study. The t result means there is a significant difference between the Buzzy intervention group and the control group in terms of the procedural selfreported pain measured with the VAS. As a point of clarification, p values are never zero in a study. There is always some chance of error.
4. The null hypothesis is: There is no difference in observerreported procedural anxiety levels between the Buzzy intervention and the control groups for schoolage children. The t = −6.745 for observerreported procedural anxiety levels, p = 0.000, which is less than α = 0.05 set for this study. Since this study result was statistically significant, the null hypothesis was rejected.
5. The t = −1.309 for BMI. The nonsignificant p = .192 for BMI is greater than α = 0.05 set for this study. The nonsignificant result means there is no statistically significant difference between the Buzzy intervention and control groups for BMI. The two groups need to be similar for demographic variables to decrease the potential for error and increase the likelihood that the results are an accurate reflection of reality.
6. The conduct of multiple ttests causes an increased risk for Type I errors. If only one ttest is conducted on study data, the risk of Type I error does not increase. The Bonferroni procedure and the more stringent Tukey’s honestly significant difference (HSD), Student NewmanKeuls, or Scheffé test can be calculated to reduce the risk of a Type I error (Plichta & Kelvin, 2013; Zar, 2010).
7. The Bonferroni procedure is calculated by alpha ÷ number of ttests conducted on study variables’ data. Note that researchers do not always report all ttests conducted, especially if they were not statistically significant. The ttests conducted on demographic data are not of concern. Canbulat et al. reported the results of four ttests conducted to examine differences between the intervention and control groups for the dependent variables procedural selfreported pain with WBFS, procedural selfreported pain with VAS, parentreported anxiety levels, and observerreported anxiety levels. The Bonferroni calculation for this study: 0.05 (alpha) ÷ number of ttests conducted = 0.05 ÷ 4 = 0.0125. The new α set for the study is 0.0125.
8. Based on the Bonferroni result = 0.0125 obtained in Question 7, the t = −6.745, p = 0.000, is still significant since it is less than 0.0125.
9. The intervention and control groups were examined for differences related to the demographic variables gender, age, and BMI and the dependent variable preprocedural anxiety that might have affected the procedural pain and anxiety posttest levels in the children 7 to 12 years old. These nonsignificant results indicate the intervention and control groups were similar or equivalent for these variables at the beginning of the study. Thus, Canbulat et al. (2015) can conclude the significant differences found between the two groups for procedural pain and anxiety levels were probably due to the effects of the intervention rather than sampling error or initial group differences.
10. No, the independent samples ttest would not have been appropriate to analyze the differences in gender between the Buzzy intervention and control groups. The demographic variable gender is measured at the nominal level or categories of females and males. Thus, the χ^{2} test is the appropriate statistic for analyzing gender data (see Exercise 19). In contrast, the ttest is appropriate for analyzing data for the demographic variables age and BMI measured at the ratio level.
EXERCISE 16 Questions to Be Graded
Follow your instructor’s directions to submit your answers to the following questions for grading. Your instructor may ask you to write your answers below and submit them as a hard copy for grading. Alternatively, your instructor may ask you to use the space below for notes and submit your answers online at http://evolve.elsevier.com/Grove/Statistics/ under “Questions to Be Graded.”
Name: _______________________________________________________ Class: _____________________
Date: ___________________________________________________________________________________
1. What do degrees of freedom (df) mean? Canbulat et al. (2015) did not provide the dfs in their study. Why is it important to know the df for a t ratio? Using the df formula, calculate the df for this study.
2. What are the means and standard deviations (SDs) for age for the Buzzy intervention and control groups? What statistical analysis is conducted to determine the difference in means for age for the two groups? Was this an appropriate analysis technique? Provide a rationale for your answer.
3. What are the t value and p value for age? What do these results mean?
4. What are the assumptions for conducting the independent samples ttest?
5. Are the groups in this study independent or dependent? Provide a rationale for your answer.
6. What is the null hypothesis for procedural selfreported pain measured with the Wong Baker Faces Scale (WBFS) for the two groups? Was this null hypothesis accepted or rejected in this study? Provide a rationale for your answer.
7. Should a Bonferroni procedure be conducted in this study? Provide a rationale for your answer.
8. What variable has a result of t = −6.135, p = 0.000? What does the result mean?
9. In your opinion, is it an expected or unexpected finding that both t values on Table 2 were found to be statistically significant. Provide a rationale for your answer.
10. Describe one potential clinical benefit for pediatric patients to receive the Buzzy intervention that combined cold and vibration
Understanding Paired or Dependent Samples tTest
Statistical Technique in Review
The paired or dependent samples ttest is a parametric statistical procedure calculated to determine differences between two sets of repeated measures data from one group of people. The scores used in the analysis might be obtained from the same subjects under different conditions, such as the one group pretest–posttest design. With this type of design, a single group of subjects experiences the pretest, treatment, and posttest. Subjects are referred to as serving as their own control during the pretest, which is then compared with the posttest scores following the treatment. Paired scores also result from a onegroup repeated measures design, where one group of participants is exposed to different levels of an intervention. For example, one group of participants might be exposed to two different doses of a medication and the outcomes for each participant for each dose of medication are measured, resulting in paired scores. The one group design is considered a weak quasiexperimental design because it is difficult to determine the effects of a treatment without a comparison to a separate control group (Shadish, Cook, & Campbell, 2002).
A less common type of paired groups is when the groups are matched as part of the design to ensure similarities between the two groups and thus reduce the effect of extraneous variables (Grove, Burns, & Gray, 2013; Shadish et al., 2002). For example, two groups might be matched on demographic variables such as gender, age, and severity of illness to reduce the extraneous effects of these variables on the study results. The assumptions for the paired samples ttest are as follows:
Research Article
Source
Lindseth, G. N., Coolahan, S. E., Petros, T. V., & Lindseth, P. D. (2014). Neurobehavioral effects of aspartame consumption. Research in Nursing & Health, 37(3), 185–193.
Introduction
Despite the widespread use of the artificial sweetener aspartame in drinks and food, there are concern and controversy about the mixed research evidence on its neurobehavioral 172effects. Thus Lindseth and colleagues (2014) conducted a onegroup repeated measures design to determine the neurobehavioral effects of consuming both low and highaspartame diets in a sample of 28 college students. “The participants served as their own controls. . . . A random assignment of the diets was used to avoid an error of variance for possible systematic effects of order” (Lindseth et al., 2014, p. 187). “Healthy adults who consumed a studyprepared highaspartame diet (25 mg/kg body weight/day) for 8 days and a lowaspartame diet (10 mg/kg body weight/day) for 8 days, with a 2week washout between the diets, were examined for withinsubject differences in cognition, depression, mood, and headache. Measures included weight of foods consumed containing aspartame, mood and depression scales, and cognitive tests for working memory and spatial orientation. When consuming highaspartame diets, participants had more irritable mood, exhibited more depression, and performed worse on spatial orientation tests. Aspartame consumption did not influence working memory. Given that the higher intake level tested here was well below the maximum acceptable daily intake level of 40–50 mg/kg body weight/day, careful consideration is warranted when consuming food products that may affect neurobehavioral health” (Lindseth et al., 2014, p. 185).
Relevant Study Results
“The mean age of the study participants was 20.8 years (SD = 2.5). The average number of years of education was 13.4 (SD = 1.0), and the mean body mass index was 24.1 (SD = 3.5). . . . Based on Vandenberg MRT scores, spatial orientation scores were significantly better for participants after their lowaspartame intake period than after their high intake period (Table 2). Two participants had clinically significant cognitive impairment after consuming highaspartame diets. . . . Participants were significantly more depressed after they consumed the highaspartame diet compared to when they consumed the lowaspartame diet (Table 2). . . . Only one participant reported a headache; no difference in headache incidence between high and lowaspartame intake periods could be established” (Lindseth et al., 2014, p. 190).
TABLE 2
WITHINSUBJECT DIFFERENCES IN NEUROBEHAVIOR SCORES AFTER HIGH AND LOW ASPARTAME INTAKE (N = 28)
Variable  M  SD  Paired tTest  p 
Spatial orientation  
Highaspartame  14.1  4.2  2.4  .03* 
Lowaspartame  16.6  4.3  
Working memory  
Highaspartame  730.0  152.7  1.5  N.S. 
Lowaspartame  761.1  201.6  
Mood (irritability)  
Highaspartame  33.4  9.0  3.4  .002** 
Lowaspartame  30.5  7.3  
Depression  
Highaspartame  36.8  7.0  3.8  .001** 
Lowaspartame  34.4  6.2 
^{*}p < .05.
^{**}p < .01.
M = Mean; SD = Standard deviation; N.S. = Nonsignificant.
Lindseth, G. N., Coolahan, S. E., Petros, T. V., & Lindseth, P. D. (2014). Neurobehavioral effects of aspartame consumption. Research in Nursing & Health, 37(3), p. 190
Study Questions
1. Are independent or dependent (paired) scores examined in this study? Provide a rationale for your answer.
2. What independent (intervention) and dependent (outcome) variables were included in this study?
3. What inferential statistical technique was calculated to examine differences in the participants when they received the highaspartame diet intervention versus the lowaspartame diet? Is this technique appropriate? Provide a rationale for your answer.
4. What statistical techniques were calculated to describe spatial orientation for the participants consuming low and highaspartame diets? Were these techniques appropriate? Provide a rationale for your answer.
5. What was the dispersion of the scores for spatial orientation for the high and lowaspartame diets? Is the dispersion of these scores similar or different? Provide a rationale for your answer.
6. What is the paired ttest value for spatial orientation between the participants’ consumption of high and lowaspartame diets? Are these results significant? Provide a rationale for your answer.
7. State the null hypothesis for spatial orientation for this study. Was this hypothesis accepted or rejected? Provide a rationale for your answer.
8. Discuss the meaning of the results regarding spatial orientation for this study. What is the clinical importance of this result? Document your answer.
9. Was there a significant difference in the participants’ reported headaches between the high and lowaspartame intake periods? What does the result indicate?
10. What additional research is needed to determine the neurobehavioral effects of aspartame consumption?
Answers to Study Questions
1. This study was conducted using one group of 28 college students who consumed both high and low aspartame diets and differences in their responses to these two diets (interventions) were examined. Lindseth et al. (2014, p. 187) stated that “the participants served as their own controls” in this study, indicating the scores from the one group are paired. In Table 2, the ttests are identified as paired ttests, which are conducted on dependent or paired samples.
2. The interventions were highaspartame diet (25 mg/kg body weight/day) and lowaspartame diet (10 mg/kg body weight/day). The dependent or outcome variables were spatial orientation, working memory, mood (irritability), depression, and headaches (see Table 2 and narrative of results).
3. Differences were examined with the paired ttest (see Table 2). This statistical technique is appropriate since the study included one group and the participants served as their own control (Plichta & Kelvin, 2013). The dependent variables were measured at least at the interval level for each subject following their consumption of high and lowaspartame diets and were then examined for differences to determine the effects of the two aspartame diets.
4. Means and standard deviations (SDs) were used to describe spatial orientation for high and lowaspartame diets. The data in the study were considered at least interval level, so means and SDs are the appropriate analysis techniques for describing the study dependent variables (Grove et al., 2013).
5. Standard deviation (SD) is a measure of dispersion that was reported in this study. Spatial orientation following a highaspartame diet had an SD = 4.2 and an SD = 4.3 for a lowaspartame diet. These SDs are very similar, indicating similar dispersions of spatial orientation scores following the two aspartame diets.
6. Paired ttest = 2.4 for spatial orientation, which is a statistically significant result since p = .03*. The single asterisk (*) directs the reader to the footnote at the bottom of the table, which identifies * p < .05. Since the study result of p = .03 is less than α = .05 set for this study, then the result is statistically significant.
7. There is no significant difference in spatial orientation scores for participants following consumption of a lowaspartame diet versus a highaspartame diet. The null hypothesis was rejected because of the significant difference found for spatial orientation (see the answer to Question 6). Significant results cause the rejection of the null hypothesis and lend support to the research hypothesis that the levels of aspartame do effect spatial orientation.
8. The researchers reported, “Based on Vandenberg MRT scores, spatial orientation scores were significantly better for participants after their lowaspartame intake period than after their high intake period (Table 2)” (Lindseth et al., 2014, p. 190). This result is clinically important since the highaspartame diet significantly reduced the participants’ spatial orientation. 176Healthcare providers need to be aware of this finding, since it is consistent with previous research, and encourage people to consume fewer diet drinks and foods with aspartame. The American Heart Association and the American Diabetic Association have provided a statement about the effects of aspartame that can be found on the National Guideline Clearinghouse website at http://www.guideline.gov/content.aspx?id=38431&search=effects+aspartame.
9. There was no significant difference in reported headaches based on the level (high or low) of aspartame diet consumed. Additional research is needed to determine if this result is an accurate reflection of reality or is due to design weaknesses, sampling or data collection errors, or chance (Grove et al., 2013).
10. Additional studies are needed with larger samples to determine the effects of aspartame in the diet. Lindseth et al. (2014) conducted a power analysis that indicated the sample size should have been at least 30 participants. Thus, the sample size was small at N = 28, which increased the potential for a Type II error. Diets higher in aspartame (40–50 mg/kg body weight/day) should be examined for neurobehavioral effects. Longitudinal studies to examine the effects of aspartame over more than 8 days are needed. Future research needs to examine the length of washout period needed between the different levels of aspartame diets. Researchers also need to examine the measurement methods to ensure they have strong validity and reliability. Could a stronger test of working memory be used in future research?
EXERCISE 17 Questions to Be Graded
Name: _______________________________________________________ Class: _____________________
Date: ___________________________________________________________________________________
Follow your instructor’s directions to submit your answers to the following questions for grading. Your instructor may ask you to write your answers below and submit them as a hard copy for grading. Alternatively, your instructor may ask you to use the space below for notes and submit your answers online at http://evolve.elsevier.com/Grove/Statistics/ under “Questions to Be Graded.”
1. What are the assumptions for conducting a paired or dependent samples ttest in a study? Which of these assumptions do you think were met by the Lindseth et al. (2014) study?
2. In the introduction, Lindseth et al. (2014) described a “2week washout between diets.” What does this mean? Why is this important?
3. What is the paired ttest value for mood (irritability) between the participants’ consumption of high versus lowaspartame diets? Is this result statistically significant? Provide a rationale for your answer.
4. State the null hypothesis for mood (irritability) that was tested in this study. Was this hypothesis accepted or rejected? Provide a rationale for your answer.
5. Which t value in Table 2 represents the greatest relative or standardized difference between the high and lowaspartame diets? Is this t value statistically significant? Provide a rationale for your answer.
6. Discuss why the larger t values are more likely to be statistically significant.
7. Discuss the meaning of the results regarding depression for this study. What is the clinical importance of this result?
8. What is the smallest, paired ttest value in Table 2? Why do you think the smaller t values are not statistically significant?
9. Discuss the clinical importance of these study results about the consumption of aspartame. Document your answer with a relevant source.
10. Are these study findings related to the consumption of high and lowaspartame diets ready for implementation in practice? Provide a rationale for your answer.
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