Independent and Paired Samples T-Test: Assumptions, Means, and Interpretation

This article discusses the assumptions, means, and interpretation of independent and paired samples t-test with solved examples. The article covers the normality test, means, t-test value, and significance level of the tests. The impact of rehabilitation on emotional distress levels and the weaknesses of the design are also discussed. The subject is statistics for nursing research, and the course code is not mentioned. The article is relevant to college and university students studying nursing research.

Statistics
Student Name:
Instructor Name:
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22nd August 2018
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Exercise 31
1. Do the example data meet the assumptions for the independent samples t -test? Provide a
rationale for your answer.
Answer
The flowing assumptions were all met:
a. Normal distribution of the datasets.
b. Dependent variable measured on ratio or interval level.
c. Homogeneity of variance in the two groups, that is, the two groups need to have
equal variance.
d. The groups should be independent of each other.
e. A large sample size
2. If calculating by hand, draw the frequency distributions of the dependent variable, wages
earned. What is the shape of the distribution? If using SPSS, what is the result of the
Shapiro-Wilk test of normality for the dependent variable?
Answer
The shape of the distribution is bell-shaped, the results of the Shapiro-Wilk test are
shown in the table below. As can be seen, the p-value is .194 (a value greater than 5%
level of significance) which means that the null hypothesis of normality is not rejected
hence the data is normally distributed or follows a normal distribution.
Tests of Normality
Kolmogorov-Smirnova Shapiro-Wilk
Statistic df Sig. Statistic df Sig.
Support .155 20 .200* .935 20 .194
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
3. What are the means for two group’s wages earned?
Answer
The table for the comparison of the means of the two groups is given below. As can be
seen, the mean for the experimental group is $232.70 while that for the control group is
$128.40.
Group Statistics
Group N Mean
Std.
Deviation
Std. Error
Mean
Support 1 10 232.70 65.325 20.658
2 10 128.40 43.025 13.606
4. What is the independent samples t -test value?
Answer
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The value of the t-test is given as 4.217.
5. Is the t -test significant at a = 0.05? Specify how you arrived at your answer.
Answer
Yes based on the given t-value, the test is statistically significant at 55 level of
significance; this because the p-value for the test is 0.001 (a value less than 5% level of
significance).
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
Support Equal variances assumed 2.477 .133 4.217 18 .001 104.300 24.736 52.332 156.268
Equal variances not assumed 4.217 15.572 .001 104.300 24.736 51.745 156.855
6. If using SPSS, what is the exact likelihood of obtaining a t -test value at least as extreme
or as close to the one that was actually observed, assuming that the null hypothesis is
true?
Answer
The given p-value is equivalent to the likelihood of making a type 1 error. The p-value
was 0.001, implying that there is a 0.1% likelihood that the null hypothesis is true.
7. Which group earned the most money post-treatment?
Answer
The group that earned the most money post-treatment is those with the assistance
program.
8. Write your interpretation of the results as you would in an APA-formatted journal.
Answer
An independent samples t-test was performed to test whether there is significant
difference in the earnings of the Veterans with disabilities for those who participated in a
work assistance program and those who did not participate in work assistance program.
Results showed that Veterans with disabilities who participated in a work assistance
program (M = $232.70, SD = $65.32, n = 10) earned significantly more money than those
veterans with disability who did not receive work assistance (M = $128.40, SD = $43.02,
n = 10), t(18) = 4.22, p = 0.001.
9. What do the results indicate regarding the impact of the supported employment
vocational rehabilitation on wages earned?
Answer
Results showed that supported employment vocational rehabilitation has a positive effect
on the wages earned by disabled veterans. Veteran with disabilities and participate in a
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work assistance program earn on average more wages than those veterans with
disabilities but don’t participate in a work assistance program.
10. Was the sample size adequate to detect significant differences between the two groups in
this example? Provide a rationale for your answer.
Answer
The sample size for both the two groups was 20. A sample size of 20 is indeed adequate
enough to help detect statistically significant difference between the two groups in the
study. This can be seen in the observed significant difference between the mean amount
of money earned by those disabled veterans who participated in work assistance program
and those who did not participate in work assistance program.
(Grove 177-178)
Grove, Susan K., Daisha Cipher. Statistics for Nursing Research: A Workbook for Evidence-
Based Practice, 2nd Edition. Saunders, 022016. VitalBook file.
4

Exercise 32
1. Do the example data meet the assumptions for the paired samples t -test? Provide a
rationale for your answer.
Answer
a. Normal distribution of the datasets.
b. Dependent variable measured on ratio or interval level.
c. Homogeneity of variance in the two groups, that is, the two groups need to have
equal variance.
d. The groups should be independent of each other.
e. A large sample size
All the above assumptions were met for this study.
2. If calculating by hand, draw the frequency distributions of the two variables. What are the
shapes of the distributions? If using SPSS, what are the results of the Shapiro-Wilk tests
of normality for the two variables?
Answer
Using SPSS, a normality test was performed. The Shapiro-Wilk test has a p-value of
0.705 for the baseline and 0.292 for the Post-test scores. The values are both greater than
5% level of significance implying that the datasets follow a normal distribution.
Tests of Normality
Kolmogorov-Smirnova Shapiro-Wilk
Statistic df Sig. Statistic df Sig.
BaselineAffScor
e
.134 10 .200* .953 10 .705
PostAffScore .235 10 .124 .912 10 .292
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
3. What are the means for the baseline and post-treatment affective distress scores,
respectively?
Answer
The baseline mean for the study is 3.030, while the score for the post-treatment was
found to be 2.040.
4. What is the paired samples t -test value?
Answer
From the study, the value of the t-test was found to be 2.865.
5. Is the t -test significant at α = 0.05? Specify how you arrived at your answer.
Answer
The t-test is significant at α = 0.05 since the p-value for the test was found to be 0.019 (a
value less than α = 0.05).
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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
BaselineAffScore -
PostAffScore
.9900 1.0929 .3456 .2082 1.7718 2.865 9 .019
6. If using SPSS, what is the exact likelihood of obtaining a t- test value at least as extreme
as or as close to the one that was actually observed, assuming that the null hypothesis is
true?
Answer
P-value of the test is equivalent to the likelihood of type 1 error being committed. This
value was found to be 0.019, implying that there is a 1.9% likelihood of the null
hypothesis being true.
7. On average, did the affective distress scores improve or deteriorate over time? Provide a
rationale for your answer.
Answer
On average the affective distress scores were higher before than after implying that the
scores improved over time. This can be seen in the difference of the means observed
between the pre-treatment and post-treatment. The mean scores for the pre-treatment was
3.03, and the mean scores for the post-treatment was 2.04 indicating a decline in the score
ratings of distress. The above results were found to be statistically significantly different.
8. Write your interpretation of the results as you would in an APA-formatted journal.
Answer
A paired-samples t-test was performed to compare the amount of emotional distress
before treatment (pre-treatment) and after treatment (post-treatment). There was
significant difference in the amount of emotional distress for the pre-treatment and post-
treatment emotional distress, t(9) = 2.865, p = 0.019, M = 0.99, SD = 1.09.
9. What do the results indicate regarding the impact of the rehabilitation on emotional
distress levels?
Answer
The results obtained from the study suggest that rehabilitation do affect the emotional
distress levels for people who suffer injury. Specifically, our results suggest that
rehabilitation positively influences the levels of the emotional distress for the people who
suffer injury. Those people who have undergone rehabilitation tend to gain some sort of
functional control and independence of their lives and as such they lead less distressful
life with little or no frustration or anger.
10. What are the weaknesses of the design in this example?
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Answer
One of the major weakness of the design used in this study is the small sample size used.
The number of participants included in the study was small. A larger sample size is
important since it helps give a more statistical power for the study. The small sample size
could come with bias i.e. few from the same gender, age group, ethnicity or race.
(Grove 177-178)
Grove, Susan K., Daisha Cipher. Statistics for Nursing Research: A Workbook for Evidence-
Based Practice, 2nd Edition. Saunders, 022016. VitalBook file.
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