# T-Test and ANOVA

Paper outline

Part A. Dependent t-test
Exploratory Data Analysis/Hypotheses
Comparison of Means
Part B. Independent t-test
Exploratory data analysis/hypotheses
Comparison of Means
Comparison of Designs

III. Part C. ANOVA

Exploratory data analysis/hypotheses
ANOVA
Part A. Dependent t-test

Exploratory data analysis/hypotheses
An exploratory data analysis for CreativityPre and CreativityPost indicates that the mean score for creativity pre-test is 40.15, N = 40 while that of creativity post test is 43.35, N = 40. The standard deviation for pre-test scores is 8.304 whereas the standard deviation for post-test scores is 9.598 (Table 1; Figure 1; Figure 2). The minimum pre-test score is 26 whereas the minimum post-test score is 20. The maximum pre-test score was 56 while the maximum post-test score is 59. The skewness value for pre-test value for pre-test scores is .280 and the skewness for post-test scores is -.256. The kurtosis values for pre-test and post-test scores are -.992 and -.443 respectively.
H0: Participation in a creative writing course does not result in increased scores of a creativity assessment.
H1: Participation in a creative writing course results in increased scores of a creativity assessment.
Comparison of means
Table 2 and Table 3 indicate the paired samples tests for creativity tests. There is a significant difference between the means for creativity pre-test scores (M=40.15, SE = 1.313, t(39) = 2.671, p<.05) and creativity post-test scores (M= 43.35, SE= 1.158, t(39) = 2.671, p<.05). The t-test is significant (p = .011 for 2-tailed test) and this is less than .05. This implies that scores of a creativity assessment are significantly higher after participation in a creativity writing course. As such, the alternate hypothesis is accepted: Participation in a creative writing course results in increased scores of a creativity assessment. Part B. Independent t-test Exploratory data analysis/hypotheses From table 4, it is evident that the mean creativity scores for the pre-test and post-test groups is 41.75, N = 80 and a standard error of mean of 1.013. The standard deviation for the same data is 9.062. The skewness value for the creativity scores for both groups is .033 indicating that the data assumes a normal distribution. The variance for the same data is 82.114 and the data is assumed to have homogeneity of variance. The kurtosis is a negative value of -.770. Figure 3 indicates the mean creativity test scores for both pre-test and post-test scores as shown by the simple bar graphs with error bars (95% CI). The bar graphs display the mean creativity scores for post-test to be higher than for pre-test scores. H0: Participation in a creative writing course does not result in increased scores of a creativity assessment. H1: Participation in a creative writing course results in increased scores of a creativity assessment. Comparison of means The t-statistic from Table 6 is used to interpret whether the variances in the pre-test and post-test creativity scores are significantly different. From Table 6, t(78) = -1.595, p>.05. The values in the “equal variances assumed” row have been considered in this case since the t-test (Similar to Levene’s test) is greater than .05. Thus, one cannot have enough confidence to conclude that variances between the pre-test and post-test scores are equal (Field, 2009). Thus, the variances can be assumed to be equal. The t-test is non-significant (p = .115 for 2-tailed test) since this is greater than .05 and the null hypothesis cannot be merely rejected. It is therefore conclusive that participation in a creative writing course does not result in increased scores of a creativity assessment.
Comparison of designs
The first analysis was a ‘within subjects’ design whereas the second analysis was a ‘between subjects’ design. The within subjects design utilized dependent t-test analysis where the pre-test scores differed significantly from the post-test scores i.e. pre-test scores (t(39) = 2.671, p<.05) and post-test scores, t(39) = 2.671, p<.05). In this case, p = .011 for 2-tailed test which is less than .05 and thus the difference is significant. In the between subjects design, independent t-test indicted that the difference between the means of the pre-test and post-test scores was non-significant, t(78) = -1.595, p>.05. In the between subjects design, p = .115 for 2-tailed test and this is greater than .05. In the within subjects design, the means for pretest and post-test scores are M=40.15, SE = 1.313 and M= 43.35, SE= 1.158 respectively. The mean for between subjects design is 41.75, SE = 1.013. The bar graph for the between subjects designs clearly show that the mean creativity scores for the post-test are greater than those for pre-test.
From the within subjects design, there was a significant difference between the pre-test and post-test creativity scores. As such, the within subjects design accepted the alternate hypothesis that participation in a creative writing course results in increased scores of a creativity assessment. This means that students who participated in creativity writing course were expected to record higher scores in a creativity assessment compared to those who did not participate in the writing course. On the other hand, the between subjects design portrayed that the difference between the pre-test and post-test scores was non-significant. In other words, the null hypothesis was accepted that participation in a creative writing course does not result in increased scores of a creativity assessment. This implies that creativity test scores were almost similar regardless of whether the student participated in a creativity writing course or not. The two tests therefore resulted into different findings.
The within subjects design was expected to display very minor differences between the means if any since the subjects of the study were from the same population. In other words, the mean for pre-test scores was expected to be roughly equal to the mean for post-test scores but this was not the case. It is therefore a surprise that the null hypothesis was not accepted yet it should have been true for two samples of data obtained from the same population. In the between subjects design, it was expected that there would be a difference between the means for the two groups since the two samples were merged and the assumption was that the samples came form the same population. It was therefore expected that the null hypothesis would be accepted. This is because despite not matching the test scores against the correct participants, the mean scores for the samples are expected to remain the same in either the pre-test or the post-test scores (reordering the test scores does not affect the mean as long as the size of the samples is the same).
From this activity, I have appreciated that comparing means for within subjects is a good way of testing hypothesis as it is possible to identify differences in the means. This therefore provides a basis for accepting or not accepting either the null or alternate hypothesis. In addition, the between subjects design has portrayed that it can be utilized in cases where the sample comes from the same population despite lack of knowledge on the distribution of the data. In summary, this activity has enlightened me on the use of dependent and independent t-tests as a way of testing hypotheses.
Part C. ANOVA
Exploratory data analysis/hypotheses
For participants who were tested for blood pressure at home, the mean systolic pressure was 122.90, N = 10 with a standard error of mean of 2.243 whereas the mean diastolic pressure was 82.90, N = 10 and SE = .849. The standard deviation for systolic blood pressure for the same group was 7.094 whereas the standard deviation for diastolic blood pressure was 2.685 (Table 7). The skewness for systolic and diastolic blood pressure in a home setting was .291 and .434 respectively. The kurtosis value was -.922 and -.002 for systolic and diastolic blood pressure. Table 7 also shows that the minimum systolic and diastolic blood pressures in a home setting were 113 and 79 respectively whereas the maximum systolic and diastolic blood pressures were 135 and 88 respectively.
According to Table 8, the mean systolic and diastolic blood pressures for participants tested in a doctor’s office setting were 132.60, SE = 2.647, N=10 and 83.20, SE = 1.062, N = 10 respectively. The standard deviation for the sample was 8.369 and 3.360 for systolic and diastolic blood pressure respectively. The minimum systolic and diastolic blood pressure measures taken in a doctor’s office were 120 and 78 respectively whereas the maximum systolic and diastolic blood pressure measures were 145 and 90 respectively. Table 8 also shows that skewness for systolic and diastolic blood pressures were .322 and .707 respectively.
According to Table 9, the mean systolic and diastolic blood pressure for participants tested in a classroom setting were 118.80, SE = 1.756, N=10 and 82.60, SE = .846, N = 10 respectively. The standard deviation for the sample was 5.554 and 2.675 for systolic and diastolic blood pressure respectively. The minimum systolic and diastolic blood pressure measures taken in a classroom setting were 110 and 79 respectively whereas the maximum systolic and diastolic blood pressure measures were 128 and 88 respectively. Table 9 also shows that skewness for systolic and diastolic blood pressures were .202 and .754 respectively.
Figure 4 indicates that systolic blood pressure is highest when blood pressure is measured in a doctor’s office, lower in a home setting and lowest in a classroom setting. The diastolic blood pressures were almost the same in all settings although it was slightly higher in a doctor’s office.
H0: Systolic blood pressures are not equivalent in a home setting, doctor’s office and in a classroom setting but diastolic blood pressures are equivalent in the same settings.
H1: Systolic blood pressures are equivalent in a home setting, doctor’s office and in a classroom setting but diastolic blood pressures are not equivalent in the same settings.
ANOVA
The mean systolic blood pressures for participants tested at home, in the doctor’s office and in classroom are 122.90, 132.60, and 118.80 respectively. The standard deviations for systolic blood pressures at home, in the doctor’s office and in a classroom setting are 7.094, 8.369, and 5.554 respectively (Table 10). From Table 11, it is evident that the analysis of variance shows a significant different between the means for the three groups of settings, F(2, 27) =9.964, p<.05 (Table 11). After performing post-hoc analyses to determine differences between the groups, it is evident that there is variability in the means of systolic blood pressures depending on the setting. Using the Tukey HSD test, it is clear that there is a significant difference between systolic blood pressure taken in a home setting compared to taking the blood pressure in a doctor’s office (p = .013) which is less than .05. However, there is no significant difference when systolic blood pressure at home and in a classroom setting (p =.412) which is greater than .05 (Table 12). Table 12 also indicates a significant difference between systolic blood pressure taken in a doctor’s office compared to tests done in a classroom setting (p =.001) which is less than .05. The Games-Howell test also confirms the same findings as indicated by a significant difference between systolic blood pressures measures taken at home against those taken at the doctor’s office (p = .031). The systolic blood pressures taken at home have no significant difference with those taken in a classroom setting (p = .344). there is a significant difference between systolic blood pressures taken in a doctor’s office compared to systolic blood pressures taken in a classroom setting (p =.001) (Table 12). Taking home setting as a control, Dunnett’s test indicates a significant difference between systolic blood pressures taken at a doctor’s office compared to those taken in a home setting (p =.010) but no significant difference between classroom and home setting (p = .342). The effect size r for the systolic blood pressures is calculated as: R2 = SSM/SST (Field, 2009) Where SSM is between-groups effect size and SST is total amount of variance in data R2 =1004.467/2365.367 r= 0.652 Converting the r into percentage, it implies that taking systolic blood pressure at a doctor’s office causes a 65% change in systolic blood pressure compared to taking the same measure in a home setting while all other variables are held constant. The mean diastolic blood pressures for participants tested at home, in the doctor’s office and in classroom are 82.90, 83.20 and 82.60 respectively. The standard deviations for diastolic blood pressures at home, in the doctor’s office and in a classroom setting are 2.685, 3.360 and 2.675 respectively (Table 13). From Table 13, it is evident that the analysis of variance shows that there is no significant different between the means for the three groups of settings, F(2, 27) = .105, p>.05 (Table 14). After performing post-hoc analyses to determine differences between the groups, it is evident that there is no significant variability in the means of diastolic blood pressures depending on the setting.
Using the Tukey HSD test, it is clear that there is no significant difference between diastolic blood pressure taken in a home setting compared to taking the blood pressure in a doctor’s office (p = .971) which is greater than .05. This is the same for diastolic blood pressures taken at home compared to those taken in a classroom setting (Table 15). Table 15 also indicates a non-significant difference between diastolic blood pressure taken in a doctor’s office compared to tests done in a classroom setting (p =.891) which is greater than .05. The Games-Howell test also confirms the same findings as indicated by a non-significant difference between diastolic blood pressures taken at home against those taken at the doctor’s office (p = .974). The diastolic blood pressures taken at home have no significant difference with those taken in a classroom setting (p = .966). There is also no significant difference between diastolic blood pressures taken in a doctor’s office compared to diastolic blood pressures taken in a classroom setting (p =.899) (Table 15). Taking home setting as a control, Dunnett’s test indicates a non- significant difference between diastolic blood pressures taken at a doctor’s office compared to those taken in a home setting (p =.962) but no significant difference between classroom and home setting (p = .962).
The effect size r for the diastolic blood pressures is calculated as:
r2 = SSM/SST (Field, 2009)
Where SSM is between-groups effect size an SST is total amount of variance in data
r2 =1.8/232.7
r= .088
Converting the r into percentage, it implies that taking diastolic blood pressure at a doctor’s office causes 8.8% change in diastolic blood pressure compared to taking the same measure in a home setting while all other variables are held constant.
In summary, the null hypothesis “systolic blood pressures are not equivalent in a home setting, doctor’s office and in a classroom setting but diastolic blood pressures are equivalent in the same settings” is accepted.

Reference
Field, A. (2009). Discovering statistics using SPSS (3rd ed.). Los Angeles: Sage. ISBN: 9781847879073.
Appendix

Table 1: Descriptive Statistics for Creativity Pre-test and Post-test Scores

Statistics

Creativity pre-test
Creativity post-test

N
Valid
40
40

Missing
0
0

Mean
40.15
43.35

Std. Error of Mean
1.313
1.518

Median
38.00
44.00

Mode
38
51

Std. Deviation
8.304
9.598

Variance
68.951
92.131

Skewness
.280
-.256

Std. Error of Skewness
.374
.374

Kurtosis
-.992
-.443

Std. Error of Kurtosis
.733
.733

Minimum
26
20

Maximum
56
59

Table 2: Paired Sample Statistics for Creativity Pre-Test and Post-Test Scores

Paired Samples Statistics

Mean
N
Std. Deviation
Std. Error Mean

Pair 1
Creativity pre-test
40.15
40
8.304
1.313

Creativity post-test
43.35
40
9.598
1.518

Table 3: Dependent t-test for Creativity Pre-Test and Post-Test Scores

Paired Samples Test

Paired Differences
t
df
Sig. (2-tailed)

Mean
Std. Deviation
Std. Error Mean
95% Confidence Interval of the Difference

Lower
Upper

Pair 1
Creativity pre-test – Creativity post-test
-3.200
7.576
1.198
-5.623
-.777
-2.671
39
.011

Table 4: Descriptive statistics for Creativity Test Scores

Statistics

creativity test score

N
Valid
80

Missing
0

Mean
41.75

Std. Error of Mean
1.013

Median
41.00

Mode
51

Std. Deviation
9.062

Variance
82.114

Skewness
.033

Std. Error of Skewness
.269

Kurtosis
-.770

Std. Error of Kurtosis
.532

Minimum
20

Maximum
59

Table 5: Group Statistics for Creativity Test Scores

Group Statistics

pre-test and post-test group
N
Mean
Std. Deviation
Std. Error Mean

creativity test score
Pre-test scores
40
40.15
8.304
1.313

Post-test scores
40
43.35
9.598
1.518

Table 6: Independent t-test for Creativity Test Scores

Independent Samples Test

Levene’s Test for Equality of Variances
t-test for Equality of Means

F
Sig.
t
df
Sig. (2-tailed)
Mean Difference
Std. Error Difference
95% Confidence Interval of the Difference

Lower
Upper

creativity test score
Equal variances assumed
.632
.429
-1.595
78
.115
-3.200
2.007
-7.195
.795

Equal variances not assumed

-1.595
76.418
.115
-3.200
2.007
-7.196
.796

Table 7: Descriptives for Blood Pressure Test in a Home Setting

Statisticsa

Systolic Blood Pressure
Diastolic Blood Pressure

N
Valid
10
10

Missing
0
0

Mean
122.90
82.90

Std. Error of Mean
2.243
.849

Std. Deviation
7.094
2.685

Variance
50.322
7.211

Skewness
.291
.434

Std. Error of Skewness
.687
.687

Kurtosis
-.922
-.002

Std. Error of Kurtosis
1.334
1.334

Minimum
113
79

Maximum
135
88

Setting = Home (control)

Table 8: Descriptives for Blood Pressure Test in a Doctor’s Office Setting

Statisticsa

Systolic Blood Pressure
Diastolic Blood Pressure

N
Valid
10
10

Missing
0
0

Mean
132.60
83.20

Std. Error of Mean
2.647
1.062

Std. Deviation
8.369
3.360

Variance
70.044
11.289

Skewness
.322
.707

Std. Error of Skewness
.687
.687

Kurtosis
-.824
.873

Std. Error of Kurtosis
1.334
1.334

Minimum
120
78

Maximum
145
90

Setting = Doctor’s office

Table 9: Descriptives for Blood Pressure Test in a Classroom Setting

Statisticsa

Systolic Blood Pressure
Diastolic Blood Pressure

N
Valid
10
10

Missing
0
0

Mean
118.80
82.60

Std. Error of Mean
1.756
.846

Std. Deviation
5.554
2.675

Variance
30.844
7.156

Skewness
.202
.754

Std. Error of Skewness
.687
.687

Kurtosis
-.369
.427

Std. Error of Kurtosis
1.334
1.334

Minimum
110
79

Maximum
128
88

Setting = Classroom

Table 10: Descriptives for Systolic Blood Pressures at Home, in the Doctor’s Office and in a Classroom Setting

Descriptives

Systolic Blood Pressure

N
Mean
Std. Deviation
Std. Error
95% Confidence Interval for Mean
Minimum
Maximum

Lower Bound
Upper Bound

Home (control)
10
122.90
7.094
2.243
117.83
127.97
113
135

Doctor’s office
10
132.60
8.369
2.647
126.61
138.59
120
145

Classroom
10
118.80
5.554
1.756
114.83
122.77
110
128

Total
30
124.77
9.031
1.649
121.39
128.14
110
145

Table 11: One-way ANOVA for Systolic Blood Pressure

ANOVA

Systolic Blood Pressure

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

Table 12: Post-hoc Tests for Systolic Blood Pressure (Tukey-HSD, Games-Howell and Dunnett’s Test)

Multiple Comparisons

Dependent Variable:Systolic Blood Pressure

(I) Setting
(J) Setting
Mean Difference (I-J)
Std. Error
Sig.
95% Confidence Interval

Lower Bound
Upper Bound

Tukey HSD
Home (control)
Doctor’s office
-9.700*
3.175
.013
-17.57
-1.83

Classroom
4.100
3.175
.412
-3.77
11.97

Doctor’s office
Home (control)
9.700*
3.175
.013
1.83
17.57

Classroom
13.800*
3.175
.001
5.93
21.67

Classroom
Home (control)
-4.100
3.175
.412
-11.97
3.77

Doctor’s office
-13.800*
3.175
.001
-21.67
-5.93

Games-Howell
Home (control)
Doctor’s office
-9.700*
3.469
.031
-18.58
-.82

Classroom
4.100
2.849
.344
-3.21
11.41

Doctor’s office
Home (control)
9.700*
3.469
.031
.82
18.58

Classroom
13.800*
3.176
.001
5.59
22.01

Classroom
Home (control)
-4.100
2.849
.344
-11.41
3.21

Doctor’s office
-13.800*
3.176
.001
-22.01
-5.59

Dunnett t (2-sided)a
Doctor’s office
Home (control)
9.700*
3.175
.010
2.29
17.11

Classroom
Home (control)
-4.100
3.175
.342
-11.51
3.31

*. The mean difference is significant at the 0.05 level.

Dunnett t-tests treat one group as a control, and compare all other groups against it.

Table 13: Descriptives for Diastolic Blood Pressures at Home, in the Doctor’s Office and in a Classroom Setting

Descriptives

Diastolic Blood Pressure

N
Mean
Std. Deviation
Std. Error
95% Confidence Interval for Mean
Minimum
Maximum

Lower Bound
Upper Bound

Home (control)
10
82.90
2.685
.849
80.98
84.82
79
88

Doctor’s office
10
83.20
3.360
1.062
80.80
85.60
78
90

Classroom
10
82.60
2.675
.846
80.69
84.51
79
88

Total
30
82.90
2.833
.517
81.84
83.96
78
90

Table 14: One-way ANOVA for Diastolic Blood Pressure

ANOVA

Diastolic Blood Pressure

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

Table 15: Post-hoc Tests for Diastolic Blood Pressure (Tukey-HSD, Games-Howell and Dunnett’s Test)

Multiple Comparisons

Dependent Variable:Diastolic Blood Pressure

(I) Setting
(J) Setting
Mean Difference (I-J)
Std. Error
Sig.
95% Confidence Interval

Lower Bound
Upper Bound

Tukey HSD
Home (control)
Doctor’s office
-.300
1.308
.971
-3.54
2.94

Classroom
.300
1.308
.971
-2.94
3.54

Doctor’s office
Home (control)
.300
1.308
.971
-2.94
3.54

Classroom
.600
1.308
.891
-2.64
3.84

Classroom
Home (control)
-.300
1.308
.971
-3.54
2.94

Doctor’s office
-.600
1.308
.891
-3.84
2.64

Games-Howell
Home (control)
Doctor’s office
-.300
1.360
.974
-3.79
3.19

Classroom
.300
1.199
.966
-2.76
3.36

Doctor’s office
Home (control)
.300
1.360
.974
-3.19
3.79

Classroom
.600
1.358
.899
-2.88
4.08

Classroom
Home (control)
-.300
1.199
.966
-3.36
2.76

Doctor’s office
-.600
1.358
.899
-4.08
2.88

Dunnett t (2-sided)a
Doctor’s office
Home (control)
.300
1.308
.962
-2.75
3.35

Classroom
Home (control)
-.300
1.308
.962
-3.35
2.75

Dunnett t-tests treat one group as a control, and compare all other groups against it.

Figure 1: A histogram showing the distribution of creativity-pretest scores

Figure 2: A histogram showing the distribution of creativity post-test scores

Figure 3: Simple bar graphs for pre-test and post-test scores
Figure 4: Clustered bar graphs for systolic and diastolic blood pressures in three settings (home, doctor’s office and classroom)

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