Independent Samples t Test: Hypothesis Testing
Independent sample t test was used to examine the difference in means for Group A prior to Training Scores and Group B Revised Training Scores. The null and alternative hypotheses are as shown below:
Ho1: There is no statistically significant difference in mean values between Group A prior to Training Scores and Group B Revised Training Scores.
Ha1: There is a statistically significant difference in mean values between Group A prior to Training Scores and Group B Revised Training Scores.
Data is analyzed using Excel Data Analysis Toolpak and the results are as shown in table 1.
Table 1: Independent Sample t test
According to the analysis, Group B Revised Training Scores had a higher mean of 84.77 compared to the mean of Group A Prior Training Scores which was 69.79. The t-statistics t(87) = -9.67, p = 1.93983E-15 which is less than .05 (Warner, 2020). This implies that we reject the null hypothesis and conclude that there is a statistically significant difference in mean values between Group A prior to Training Scores and Group B Revised Training Scores.
Dependent Samples (Paired Samples) t Test: Hypothesis Testing
Paired samples t test was used to examine the difference in means for employees’ blood levels before exposure to lead and the same employees’ blood levels after exposure to lead. In this case, the variables used are Pre-Exposure μg/dL and Post-Exposure μg/dL. The null and alternative hypotheses will be given as follows:
Ho2: There is no statistically significant difference in mean values between employees’ Pre-Exposure μg/dL and Post-Exposure μg/dL.
Ha2: There is a statistically significant difference in mean values between employees’ Pre-Exposure μg/dL and Post-Exposure μg/dL.
The output for the analysis is as shown in table 2 below;
Table 2: Paired sample t test
The results shows that the mean for employees’ Post-Exposure μg/dL was slightly higher (33.29) compared to mean for employees’ Pre-Exposure μg/dL which was 32.86 μg/dL. The t test statistics t (48) = -1.93, p = 0.06 which is greater than 0.05 implying we fail to reject the null hypothesis and conclude that there is no statistically significant difference in mean values between employees’ Pre-Exposure μg/dL and Post-Exposure μg/dL (Ross, & Willson, 2017).
ANOVA: Hypothesis Testing
One-Way ANOVA was used to examine the difference in means for 4 consulting project return on investments in percentages. The 4 projects are A=Air, B=Soil, C=Water and D=Training. The null and alternative hypotheses will be given as follows:
Ho3: There is no statistically significant difference in mean return on investments between Air, Soil, Water and Training projects.
Ha3: There is a statistically significant difference in mean return on investments between Air, Soil, Water and Training projects.
The output is as shown in table 3.
Table 3: One-Way ANOVA
The results shows that the mean of return on investment for the soil project was higher (9.1%), followed by the mean of return on investment for air project (8.9%), then water project (7%) and the least was training project (5.4%).
The ANOVA results shows that differences were statistically significant where F (3, 76) = 11.92, p = 1.76E-06 which is less than 0.05 implying that we reject the null hypothesis and conclude that there is a statistically significant difference in mean return on investments between Air, Soil, Water and Training projects (Kim, 2017).
References
Warner, R. M. (2020). Applied statistics II: Multivariable and multivariate techniques. SAGE Publications, Incorporated.
Ross, A., & Willson, V. L. (2017). Paired samples T-test. In Basic and advanced statistical tests (pp. 17-19). Brill Sense.
Kim, T. K. (2017). Understanding one-way ANOVA using conceptual figures. Korean journal of anesthesiology, 70(1), 22.
t-Test: Two-Sample Assuming Unequal Variances
Group A Prior
Training Scores
Group B Revised
Training Scores
Mean69.7903225884.77419355
Variance122.00449526.96456901
Observations6262
Hypothesized Mean Difference0
df87
t Stat-9.666557191
P(T<=t) one-tail9.69914E-16
t Critical one-tail1.662557349
P(T<=t) two-tail1.93983E-15
t Critical two-tail1.987608282
t-Test: Paired Two Sample for Means
Pre-Exposure
μg/dL
Post-Exposure
μg/dL
Mean32.8571428633.28571429
Variance150.4583333155.5
Observations4949
Pearson Correlation0.992236043
Hypothesized Mean Difference0
df48
t Stat-1.929802563
P(T<=t) one-tail0.029776357
t Critical one-tail1.677224196
P(T<=t) two-tail0.059552714
t Critical two-tail2.010634758
Anova: Single Factor
SUMMARY
GroupsCountSumAverageVariance
A = Air201788.99.357895
B = Soil201829.13.042105
C = Water2014076.631579
D = Training201085.41.410526
ANOVA
Source of VariationSSdfMSFP-valueF crit
Between Groups182.8360.9333311.92311.76E-062.724944
Within Groups388.4765.110526
Total571.279