Paired Sample T-test And Independent Sample T-test: An Analysis

1. Table 1 presents the descriptive statistics for the Pre-test and Post-test scores of the creative writing course (Christodoulides & Christodoulides, 2017).

Table 1: Descriptive Statistics Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper
	N	Mean	Std. Deviation
CreativePre Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper	40	40.15	8.30
CreativePost	40	43.35	9.60
Valid N (listwise)	40

Descriptive Statistics

The average and standard deviation of pre-test score of creative writing is 40.15 and 8.30.  The average and standard deviation of post-test score of creative writing is 43.35 and 9.60 respectively. 

The comparative distribution of Creativity in Pre-test and Creativity in Post-test indicates that overall values are greater for Creativity in post-test scores. The score is higher for maximum and minimum values for post-test score.   

Null Hypothesis (H₀): Participation in the creative writing course does not produce any difference in the mean pre and post-test scores. 

Alternate Hypothesis (H_A): Participation in the creative writing course does produce difference in the mean pre and post-test scores (Anderson et al., 2014). 

1. Table 2 presents the results of the paired samples t-test (Black, 2016)

Table 2: Paired Samples Test
	Pair 1
CreativePre – CreativePost
Paired Differences	Mean	-3.20000
Std. Deviation	7.57594
Std. Error Mean	1.19786
95% Confidence Interval of the Difference	Lower	-5.62290
Upper	-.77710
t	-2.671
df	39
Sig. (2-tailed)	.011

The paired sample t-test is applicable when we would like to test the averages of two similar types of variables having so significant relevance between them.

The t-statistic for paired two samples is –,

Where, X₁bar = average of first sample, X₂bar = average of second sample, S² = sample variance, n = sample size, t= Student t-statistic with (n-1) degrees of freedom (Francis, 2014).

The confidence interval of the t-statistic is found to be –

(X₁bar – X₂bar) ± t *    or, equivalently, [(X₁bar – X₂bar) ± t * SE(X₁bar – X₂bar)]

The paired difference is found to be (-3.2) and t-statistic is calculated as (-2.671) with degrees of freedom 39. The p-value of significant t-statistic is found to be 0.11. It is greater than 0.05. Therefore, null hypothesis of equality of means is accepted at 5% level of significance.

Hence, we can infer that we are 95% confident that average scores of pre-test and post-test Creativity are equal.

40 students were asked to participate in a creative writing course. Prior to the start of the course they were administered a test whose score was stored as “CreativePre.” At the end of the course the students were again administered a test, the scores was stored as “CreativePost.” The scores of the participants was stored individually. To assess the importance of the creative writing course a paired sample t-test was done.

The paired-samples t-test shows that the mean of Creativity Pre-test (M = 40.15, SD = 8.30) and Creativity Post-test (M = 43.35, SD = 9.60), t(39) = -2.671, p = .011. On an average Creativity Pre-test was about 3.200 points lower than Creativity Post-test score (Berenson, Levine & Krehbiel, 2012).

A data set was created with the scores.

1. The paired samples t-test is used to judge the impact of creative writing course. The pre-test and post-test scores of individual participants are compared. The mean test scores of the participants are compared.

On the other hand, the independent sample t-test is used to judge the impact of creative writing of unrelated groups. The groups are test scores prior and post taking the course. 

Table 3: Group Statistics
	Grouping	N	Mean	Std. Deviation	Std. Error Mean
TestScore	Pre	40	40.15	8.30	1.31
Post	40	43.35	9.60	1.52

Paired Sample T-test Results

Like mean and standard deviation, standard error of 40 samples is also greater for Post-test (1.52) than Pre-test (1.31). 

The Null and Alternate Hypothesis of independent samples t-test may be written as:

Null Hypothesis (H₀): Participation in the creative writing course does not produce any difference in the mean pre and post-test scores. 

Alternate Hypothesis (H_A): Participation in the creative writing course does produce difference in the mean pre and post-test scores. 

Table 4: Independent Samples Test
	TestScore
Equal variances assumed	Equal variances not assumed
Levene’s Test for Equality of Variances	F	.632
Sig.	.429
t-test for Equality of Means	t	-1.595	-1.595
df	78	76.418
Sig. (2-tailed)	.115	.115
Mean Difference	-3.20000	-3.20000
Std. Error Difference	2.00675	2.00675
95% Confidence Interval of the Difference	Lower	-7.19514	-7.19644
Upper	.79514	.79644

Part 3

The Levene’s F-test is applicable for relating test score of equal variances assumed and not assumed. The F-test is calculated as 0.632 with p-value 0.429. The p-value is greater than 0.05. Therefore, we cannot reject the null hypothesis of equality of variance at 5% level of significance (George & Mallery, 2016).

The t-statistic for independent sample t-test is given as – 

Where,  = 

The t-test for equality of means is given as (-1.595) with degrees of freedom 78 when equal variances is assumed. The significant t-statistic is (0.115) which is greater than 0.05. Hence, from this perspective also we accept the null hypothesis.

Therefore, it is 95% evident that the averages of these two variables are equal.

We have used the same data set for analysing both a between and within subject’s design. The both t-tests that are two sample paired t-test and two samples paired t-test refer the same conclusion. In this analysis, both types of tests, it was recorded that the averages of pre-test and post-test are equal.

When identification number mismatches pre-test and post-test scores, then we applied independent samples t-test with the help of 40 pre-test scores and 40 post-test scores. For comparing pre-test and post-test scores, we applied between subject’s design rather than within subject’s design. Both the samples are imputed vertically and incorporated independent sample t-test. The between–subjects rather than within-subjects design is useful to take total 80 samples at a time. The within subject design is helpful for commonly in repeated measure analysis. The within subject design helps to measure how much an individual in the sample tends to vary in different observation. In other words, it is the mean of variation for the average individual case in the specified sample.   

1. The Analysis of variance technique helps to find the differences of mean scores between Systolic blood pressure and Diastolic blood pressure in Home, Doctor’s Office and Classroom setting. 

Descriptive Statistics
Setting	N	Mean	Std. Deviation
Home	SystolicBP	10	122.9000	7.09382
Valid N (listwise)	10
Doctor’s Office	SystolicBP	10	132.6000	8.36926
Valid N (listwise)	10
Classroom Setting	SystolicBP	10	118.8000	5.55378
Valid N (listwise)	10

The descriptive statistic indicates that systolic blood pressure is higher for Doctor’s Office followed by Home. The pressure is least in classroom setting. All the variables show same indication in both average and standard deviation values.

Independent Sample T-test Results

Descriptive Statistics
Setting	N	Mean	Std. Deviation
Home	DiastolicBP	10	82.9000	2.68535
Valid N (listwise)	10
Doctor’s Office	DiastolicBP	10	83.2000	3.35989
Valid N (listwise)	10
Classroom Setting	DiastolicBP	10	82.6000	2.67499
Valid N (listwise)	10

The descriptive statistic refers that Diastolic blood pressure has mean and standard deviation values in Doctor’s office followed by Home. The average and standard deviation are least in Classroom setting.  

The graph shows that Overall distribution of Systolic blood pressure is highest in Doctor’s office and lowest in Classroom setting.

The distributions of Diastolic blood pressures are almost equal in all the three settings that are Home, Doctor’s Office and Classroom settings.

The Null and Alternate Hypothesis may be written as:

Null Hypothesis (H₀): The averages of systolic blood pressures in all the three settings are equal that is µ₁ = µ₂ =µ₃.

Alternative Hypothesis (H_A): There exists at least one equality in the averages of all the three settings are systolic blood pressures.

Null Hypothesis (H₀): The averages of diastolic blood pressures in all the three settings are equal that is µ₁ = µ₂ =µ₃.

Alternative Hypothesis (H_A): There exists at least one equality in the averages of all the three settings are diastolic blood pressures.

1. We incorporated two ANOVA tables. One is calculated as Systolic BP is dependent variable and other is calculated as Diastolic BP, a dependent variable. 

ANOVA
SystolicBP
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	1004.467	2	502.233	9.964	.001
Within Groups	1360.900	27	50.404
Total	2365.367	29

For Systolic BP value of F-statistic is 9.964 with significant p-value 0.001. The p-value of F-statistic is less than 0.05. Therefore, we reject the null hypothesis at 5% level of significance.

Hence, it is 95% evident that the average values of Systolic blood pressures in all the three setting that are Home, Classroom setting and Doctors office are unequal. 

ANOVA
DiastolicBP
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	1.800	2	.900	.105	.900
Within Groups	230.900	27	8.552
Total	232.700	29

For Diastolic BP value of F-statistic is 0.105 with significant p-value 0.900. The p-value of F-statistic is greater than 0.05. Therefore, we accept the null hypothesis at 5% level of significance. Hence, it is 95% evident that the average values of Diastolic blood pressures in all the three setting that are Home, Classroom setting and Doctors office are equal.

The Sum of squares within groups and between groups are significantly higher in case of ANOVA test of Systolic BP. On the other hand, sum of squares between groups is insignificant than within group SSE in the ANOVA table of Diastolic blood pressure (Field, 2013). 

Multiple Comparisons
Dependent Variable:   SystolicBP
Bonferroni
(I) Setting	(J) Setting	Mean Difference (I-J)	Std. Error	Sig.	95% Confidence Interval
Lower Bound	Upper Bound
Home	Doctor’s Office	-9.70000*	3.17502	.015	-17.8041	-1.5959
Classroom Setting	4.10000	3.17502	.623	-4.0041	12.2041
Doctor’s Office	Home	9.70000*	3.17502	.015	1.5959	17.8041
Classroom Setting	13.80000*	3.17502	.001	5.6959	21.9041
Classroom Setting	Home	-4.10000	3.17502	.623	-12.2041	4.0041
Doctor’s Office	-13.80000*	3.17502	.001	-21.9041	-5.6959
*. The mean difference is significant at the 0.05 level.

The table of “multiple comparison” refers that Using Systolic blood pressure as dependent variable; we calculated different types of differences between two types of settings. The first setting involves Home, Doctor’s Office and Classroom setting (McCormick, 2017). The second setting includes paired categorical settings in each of the three cases. The p-value between Home and classroom setting is 0.623 (>0.05). Therefore, the systolic BP between home and classroom setting are equal. The p-value between Doctor’s office and classroom setting is 0.001 (<0.005). Therefore, the systolic BP between Doctor’s office and Classroom setting are not equal. Further, the p-value between Home and Doctor’s Office is 0.015 (<0.05). Hence, the systolic BP between home and Doctor’s office are not equal.

References

Anderson, D. R., Sweeney, D. J., Williams, T. A., Camm, J. D., & Cochran, J. J. (2014). Essentials of statistics for business and economics. Cengage Learning.

Berenson, M. L., Levine, D. M., & Krehbiel, T. C. (2012). Basic business statistics. Upper Saddle River, NJ: Prentice Hall.

Black, K. (2016). Business statistics: Contemporary decision making. John Wiley & Sons.

Christodoulides, C., & Christodoulides, G. (2017). Analysis and Presentation of Experimental Results. Springer

Field, A. (2013). Discovering statistics using IBM SPSS statistics. Sage.

Francis, A. (2014). Business mathematics and statistics. Cengage Learning EMEA.

George, D., & Mallery, P. (2016). IBM SPSS Statistics 23 step by step: A simple guide and reference. Routledge.

McCormick, K., Salcedo, J., Peck, J., & Wheeler, A. (2017). SPSS Statistics for data analysis and visualization. John Wiley & Sons.