Download Introduction to T-tests

DATA ANALYSIS AND APPLICATION: T-tests
Data Analysis and Application: T-tests
Heather A. Lacharite
Capella University
1
DATA ANALYSIS AND APPLICATION: T-tests
2
Introduction to T-tests
A rudimentary inferential statistic that is frequently used in psychological study is called
the t-test. There are always two variables when involving t-tests. There is the predictor variable
(X) and the outcome variable (Y). Along with the logic of t-tests, all inferential statistics include
independent variables of t-tests and all function under assumptions. In this paper, we will also
explore the null and alternative hypothesis for t-tests. The null hypothesis predicts no significant
difference in population means. When speaking of a directional alternative hypothesis for a t test,
the researcher will conclude that the population means vary in a precise track. In terms of a nondirectional alternative hypothesis, the research will propose that the population means vary,
however, it will not specify which mean is preeminent.
Data Description
Lacharite (2017), previously reported the description of the data as:
“Gender: Categorical variable, having two values 1 for female and 2 for male. The scale of
measurement here is nominal
GPA: It is the cumulative grade point average of students at the start of the course. The scale of
measurement here is ratio. GPA has a value of 1-4.
Total: It is the combined scores of quizzes one, two, three, four, five, and the final exam. It is a
value between 1-100, and it is an interval measurement.
Final: This is only the final score, also an interval measurement
The sample size is 105” (p. 3).
DATA ANALYSIS AND APPLICATION: T-tests
3
Per Howell (2011), an independent samples t-test compares the mean scores of two
groups on a given variable. For this assignment, descriptive statistics, visual representation of the
distribution, test of normality, as well as an independent samples t-test and test of homogeneity
will be created and computed. GPA and Gender are the variables used for this assignment.
Gender, which is measured on nominal scale of measurement is the first variable. This variable is
considered a dichotomous variable that has values of 1 and 2. GPA is the second variable yet this
variable is measured on an interval scale rather than a nominal scale.
Section 2: Testing Assumption
Some assumptions that can be made are, that the dependent variable can be considered as
normally distributed, this can be checked with the Q-Q plot. The next assumption is that the two
groups are considered independent of each other. In addition, the two groups have no effect on
the dependent variable. This can be checked by using the Levene’s Test.
Case Processing Summary
gender
Cases
Valid
N
gpa
Missing
Percent
N
Total
Percent
N
Percent
1
64
100.0%
0
.0%
64
100.0%
2
41
100.0%
0
.0%
41
100.0%
d
i
m
e
n
s
i
o
n
1
DATA ANALYSIS AND APPLICATION: T-tests
4
Descriptive
Gender
gpa
1
Statistic
Mean
95% Confidence Interval for Mean
99.33
Lower Bound
95.76
Upper Bound
102.89
5% Trimmed Mean
98.79
Median
99.00
Variance
1.779
174.039
Std. Deviation
2
Std. Error
13.192
Minimum
75
Maximum
137
Range
62
Interquartile Range
20
Skewness
.451
.322
Kurtosis
.224
.634
101.82
2.207
Mean
95% Confidence Interval for Mean
Lower Bound
97.32
Upper Bound
106.31
5% Trimmed Mean
101.41
Median
100.00
Variance
Std. Deviation
160.716
12.677
Minimum
83
Maximum
128
Range
45
Interquartile Range
20
Skewness
Kurtosis
.354
.409
-.731
.798
DATA ANALYSIS AND APPLICATION: T-tests
5
Tests of Normality
gender
Kolmogorov-Smirnova
Statistic
gpa
df
Shapiro-Wilk
Sig.
Statistic
df
Sig.
d1
.079
55
.200*
.972
55
.235
i
2
.101
33
.200*
.960
33
.261
m
e
n
s
i
o
n
1
a. Lilliefors Significance Correction
*. This is a lower bound of the true significance.
After reviewing the descriptive table output by the SPSS software, it can be determined
that both samples are drawn from normal population. Since the p-value is > than 0.05, the
samples are considered drawn from normal population. An unpaired t-test can be conducted
since both samples are independent of each other. The t-test will assess for a significant
difference of the two-sample’s means. As listed, there is a sum of 55 observations for gender 1.
Within those 55 observations there is a mean GPA of 99.33 and a standard deviation of 13.192.
In looking at Gender 2, there is a sum of 33 observations. Within the 33 observations there is a
mean GPA of 101.82 and a standard deviation of 12.677. With these calculations, one can
conclude there is a difference of -2.491 between the two samples.
The Shapiro-Wilk Test is considered a formal statistical test of normality. For this test, to
have a normal distribution results would have to indicate a value of 1.0. Anything less than 1.0
DATA ANALYSIS AND APPLICATION: T-tests
6
would represent a departure from a perfect normal shape. Thus, we can conclude for this
assignment the normal distribution is slightly departing from a perfect normal shape as the value
is indicating a 0.972 for gender 1 and 0.960 for gender 2.
In terms of the Levene’s Test for Equality of Variances, it shows if the second
assumption is satisfied. There is approximate equal variance between the two groups on the
dependent variable. The significant value will need to be 0.05 or less for either variance to be
measured as considerably diverse. A value 0.05 or greater would tell that the variances are not
considerably diverse. The Levene’s Test for Equality determined the variances show there is no
treatment effect due to the fact they are almost equal. Given the value is 0.959, it is greater than
0.05 therefore the variances are almost equal or has no treatment effect
Section 3: Research Question, Hypotheses, and Alpha Level
Per Howell (2011), an inferential procedure is utilized in statistics to make inferences
about unknown populations or scores that vary depending on the data that is utilized and the
purpose of making the inference. Regression analysis is a procedure utilized to analyze several
variables. It focuses on the relationship between a dependent variable and on or more
independent variables. Therefore, regression analysis was the inferential procedure used in this
assignment. Per Howell (2011), assuming there is not a significant difference between the
population mean and the sample mean is a null hypothesis. Furthermore, he explains that an
alternative hypothesis assumes there is in fact a significant difference. The alpha level is another
term for the level of significance. The alpha level for this assignment is .05. Below is the null
and alternative hypothesis’s in statistical form that can be tested, along with the research
question the t-test will answer:
Null Hypotheses (H o): There is no real difference between gpa of males and females.
DATA ANALYSIS AND APPLICATION: T-tests
7
Alternative Hypothesis (Ha): There is a significant difference between gpa of males and
females
Research Question: Are female GPA’s higher than males?
When an independent sample t test was completed to test the hypothesis’s, the null
hypothesis can be accepted. This is concluded due to the t-statistics value is 0.87 with 86 degrees
of freedom. Furthermore, 0.387 is the p-value for the t statistic, which is less than 0.05. With a
5% level of significance the null hypothesis is acceptable. The result being, GPA between males
and females has no real difference.
Interpretation
In using the Shapiro-Wilks and Kolmogorov-Smirnov tests, the outcomes
precisely demonstrate adequate indication that the test of normality of, GENDER and IQ, were
drawn from a normal population. Q-Q plots, box plots and the descriptive statistics, supported
this conclusion. It was detected that the two samples were autonomous of one another, which
allowed for an unpaired t-test which will test the mean of the two samples for a significant
difference. In addition, it was observed that Gender 1 has 55 subjects with a mean IQ of 99.33
and a standard deviation of 13.192, while Gender 2 has 33 subjects with a mean IQ of 101.82
and a standard deviation of 12.677. As previously stated, the difference between the two samples
IQ were observed at -2.491.
In examining the top line of the results from the independent sample t-tests, it displays
the variances are approximately equal (-.870 and -.879), but the essential point will display
variances that are not equal. However, based on the Levene’s test, results demonstrate an
approximate equal variance due to the p-value being .959. Looking at the value of t-statistics, we
see the value calculated at 0.870 with degrees of freedom calculated at 86. Given 0.870 is greater
DATA ANALYSIS AND APPLICATION: T-tests
8
than 0.05 and represents a 5% level of significance, we can accept the null hypothesis. By
accepting the null hypothesis, it is being determined there is no real difference between the IQ’s
of males and females.
Independent Samples Test
Levene's Test for
Equality of Variances
t-test for Equality of Means
95% Confidence
Interval of the
F
gpa Equal variances
assumed
Equal variances
not assumed
Sig.
t
.003
.959
df
-
Sig. (2-
Mean
Std. Error
tailed)
Difference
Difference
Difference
Lower
Upper
86
.387
-2.491
2.863 -8.183
3.201
69.665
.383
-2.491
2.835 -8.145
3.163
.870
.879
Conclusion
Overall, t-tests have been designed for independent means that are established on certain
presumptions. If the said presumptions end up being inaccurate then the results of the analysis
can be labeled flawed. When using t-tests it is important to remember it only examines the means
and does not provide information on individual scores. This could be considered a strength or
weakness depending on the context of one’s research. A limitation of the t-test is that it usually
requires a larger sample size. A sample size too small can be considered as having deficient
power (Connelly, 2011). The t-test can however be universally utilized if you have two different
groups means to be analyze. This can be with GPA and gender like in this assignment, or in a
DATA ANALYSIS AND APPLICATION: T-tests
brewery as this is where the t-test was developed. The t-test monitored the quality of stout at the
Guinness brewery in Dublin, Ireland (Connelly, 2011). Some other highlights of t-tests include,
it’s calculation is done with ease, it tells you how different the mean of one sample is from the
rest of the group, and requires little data. In my opinion, the t-test is a great statistical test die to
the fact that it is important to keep track of the conclusions of the mean rather than single scores
of the individuals, which can give researchers a more majority oriented position.
9
DATA ANALYSIS AND APPLICATION: T-tests
References
Connelly, L. M. (2011). t-tests. Medsurg Nursing, 20(6), 341. Retrieved from
http://search.proquest.com.library.capella.edu/docview/908549522?accountid=27965
Howell, D.C. (2011). Fundamental statistics for the behavioral sciences (7th ed).
Belmont, CA: Wadsworth Cengage Learning.
Lacharite, (2017). Correlations. Unpublished Manuscript. Capella University.
10