Correlation: Descriptive Statistics and Assumption Testing
Frequency distribution table.
|
Class |
Frequency- Microns |
Frequency - Mean annual sick days |
|
2 |
15 |
1 |
|
4 |
17 |
6 |
|
6 |
22 |
31 |
|
8 |
33 |
42 |
|
10 |
16 |
19 |
|
12 |
0 |
4 |
| More |
0 |
0 |
Histogram.
Descriptive statistics table.
|
microns |
mean annual sick days per employee |
|
| Mean | 5.66 | 7.13 |
| Standard Error | 0.26 | 0.19 |
| Median |
6 |
7 |
| Mode |
8 |
7 |
| Standard Deviation | 2.59 | 1.89 |
| Sample Variance | 6.73 | 3.58 |
| Kurtosis | (0.85) | 0.12 |
| Skewness | (0.37) | 0.14 |
| Range |
9.8 |
10 |
| Minimum |
0.2 |
2 |
| Maximum |
10 |
12 |
| Sum |
582.7 |
734 |
| Count |
103 |
103 |
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Measurement scale.
The two variables are quantitative in nature and assume a ratio level of measurement.
Measure of central tendency
The measures of center include the mean, median, and the mode. The mean indicates the center of the data by revealing the most typical value in a group of data. The mean microns is 5.66 while the average mean annual sick days per employee is 7.13 days. The median indicates the center by revealing the middle most value in a group of data arranged in ascending order. The median microns is 6 while the median mean annual sick days per employee is 7. The mode shows the most frequent value. The modal microns is 8 while the modal mean annual sick days per employee is 7.
Evaluation
The standard deviation shows the dispersion of data from the mean. The standard deviation for microns is 2.59 while the standard deviation for the mean annual sick days per employee is 1.89. The variance indicates the mean square deviations of the data points from the mean. The variance for microns is 6.73 while the variance for the mean annual sick days per employee is 3.58 days. The coefficient of skewness reveals the shape of the distribution of the data relative to the normal curve. The skewness for microns is -0.37 implying that the data points are negatively skewed while the skewness for mean annual sick days is 0.14 implying that the data points are slightly positively skewed.
The histogram for the two variable appears to approximately assume a normal distribution. Additionally, the frequency table indicates no presence of outliers. Lastly, the variables assume ratio scales of measurement. Assumptions for parametric statistical testing are met ( Mooi, Sarstedt, & Mooi-Reci, 2018) .
Simple Regression: Descriptive Statistics and Assumption Testing
Frequency distribution table.
|
Bin range |
Frequency |
|
40 |
5 |
|
80 |
15 |
|
120 |
27 |
|
160 |
32 |
|
200 |
51 |
|
240 |
44 |
|
280 |
28 |
|
320 |
15 |
|
360 |
6 |
| More |
0 |
Histogram.
Descriptive statistics table.
safety training expenditure |
lost time hours |
|
| Mean | 595.98 | 188.00 |
| Standard Error | 31.48 | 4.80 |
| Median |
507.772 |
190 |
| Mode |
234 |
190 |
| Standard Deviation | 470.05 | 71.73 |
| Sample Variance | 220,948.85 | 5,144.54 |
| Kurtosis | 0.44 | (0.50) |
| Skewness | 0.95 | (0.08) |
| Range |
2251.404 |
350 |
| Minimum |
20.456 |
10 |
| Maximum |
2271.86 |
360 |
| Sum |
132904.517 |
41925 |
| Count |
223 |
223 |
Measurement scale.
The values for the two variables have an origin i.e. zero and as such assume a ratio scale of measurement.
Measure of central tendency.
The measures of center include the mean, median, and the mode. The mean indicates the center of the data by revealing the most typical value in a group of data. The mean safety training expenditure is $ 595.98 while the mean lost time hours is 188 hours. The median indicates the center by revealing the middle most value in a group of data arranged in ascending order. The median safety training expenditure is 507.77 while the median lost time hours is 190. The mode shows the most frequent value. The modal safety training expenditure is 234 while the lost time hours is 190.
Evaluation.
The standard deviation shows the dispersion of data from the mean. The standard deviation for safety training expenditure is 470.05 while the standard deviation for the lost time hours is 71.73. The variance indicates the mean square deviations of the data points from the mean. The variance for safety training expenditure is 220,948.85 while the variance for the lost time hours is 5,144.54 hours . The coefficient of skewness reveals the shape of the distribution of the data relative to the normal curve. The skewness for safety training expenditure is 0.95 implying that the data points are positively skewed while the skewness for lost time hours is -0.08 implying that the data points are slightly negatively skewed.
The histogram for the dependent variable appears to approximately assume a normal distribution. Additionally, the frequency table indicates no presence of outliers. Lastly, the variables assume ratio scales of measurement. Assumptions for parametric statistical testing are met.
Multiple Regression: Descriptive Statistics and Assumption Testing
Frequency distribution table.
|
Bin range |
Frequency |
|
105 |
4 |
|
110 |
32 |
|
115 |
108 |
|
120 |
216 |
|
125 |
332 |
|
130 |
436 |
|
135 |
304 |
|
140 |
69 |
|
145 |
2 |
| More |
0 |
Histogram.
Descriptive statistics table.
|
Frequency (Hz) |
Angle in Degrees |
Chord Length |
Velocity (Meters per Second) |
Displacement |
Decibel |
|
| Mean |
2886.380572 |
6.782302063 |
0.116140053 |
50.86074518 |
0.01113988 |
124.8359 |
| Standard Error |
81.31781119 |
0.152652835 |
0.001256368 |
0.401686079 |
0.000339199 |
0.177945 |
| Median |
1600 |
5.4 |
0.1176 |
39.6 |
0.00495741 |
125.721 |
| Mode |
2000 |
0 |
0.0917 |
39.6 |
0.00529514 |
127.315 |
| Standard Deviation |
3152.573137 |
5.918128125 |
0.048707555 |
15.5727844 |
0.013150234 |
6.898657 |
| Sample Variance |
9938717.384 |
35.0242405 |
0.002372426 |
242.5116138 |
0.000172929 |
47.59146 |
| Kurtosis |
5.708685077 |
-0.412950793 |
-1.178196484 |
-1.563951274 |
2.218903124 |
-0.31419 |
| Skewness |
2.137084337 |
0.689164402 |
-0.027537436 |
0.235852414 |
1.702164556 |
-0.41895 |
| Range |
19800 |
22.2 |
0.1697 |
39.6 |
0.058010618 |
37.607 |
| Minimum |
200 |
0 |
0.03 |
31.7 |
0.000400682 |
103.38 |
| Maximum |
20000 |
22.2 |
0.1997 |
71.3 |
0.0584113 |
140.987 |
| Sum |
4338230 |
10193.8 |
174.5585 |
76443.7 |
16.74324023 |
187628.4 |
| Count |
1503 |
1503 |
1503 |
1503 |
1503 |
1503 |
Measurement scale.
All the variables; Frequency (Hz), Angle in Degrees, Chord Length, Velocity (Meters per Second), displacement, and decibel are quantitative in nature and assume a ratio scale of measurement .
Measure of central tendency.
The mean indicates the center of the data by revealing the most typical value in a group of data ( Mendenhall, Sincich, & Boudreau, 2016) . The mean for Frequency (Hz), Angle in Degrees, Chord Length, Velocity (Meters per Second), displacement, and decibel is 2886.38, 6.78, 0.116, 50.86, 0.11, and 124.85, respectively. The median indicates the center by revealing the middle most value in a group of data arranged in ascending order. The median for Frequency (Hz), Angle in Degrees, Chord Length, Velocity (Meters per Second), displacement, and decibel is 1600, 5.4, 0.1176, 39.6, 0.00496, and 125.72, respectively. The mode shows the most frequent value in a group of data. The median for Frequency (Hz), Angle in Degrees, Chord Length, Velocity (Meters per Second), displacement, and decibel is 2000, 0, 0.0917, 39.6, 0.0053, and 127.32, respectively.
Evaluation.
The standard deviation shows the dispersion of data from the mean. The standard deviation for Frequency (Hz), Angle in Degrees, Chord Length, Velocity (Meters per Second), displacement, and decibel is 3152.57, 5.91, 0.049, 15.57, 0.013, and 6.899, respectively. The variance indicates the mean square deviations of the data points from the mean. The coefficient of skewness reveals the shape of the distribution of the data relative to the normal curve. The skewness for Frequency (Hz), Angle in Degrees, Chord Length, Velocity (Meters per Second), displacement, and decibel is 2.14, 0.69, 0.028, -0.28, 1.70, and 0.419, respectively.
The histogram for the dependent variable appears to approximately assume a normal distribution. Additionally, the frequency table indicates no presence of outliers. Lastly, the variables assume ratio scales of measurement. Assumptions for parametric statistical testing are met.
Independent Samples t Test: Descriptive Statistics and Assumption Testing
Frequency distribution table.
|
Bin range |
Frequency |
|
78 |
7 |
|
81 |
10 |
|
84 |
12 |
|
87 |
14 |
|
90 |
11 |
|
93 |
5 |
|
96 |
2 |
|
99 |
1 |
| More |
0 |
Histogram.
Descriptive statistics table.
Group A Prior Training Scores |
Group B Revised Training Scores |
|
| Mean |
69.79032258 |
84.77419355 |
| Standard Error |
1.402788093 |
0.659478888 |
| Median |
70 |
85 |
| Mode |
80 |
85 |
| Standard Deviation |
11.04556449 |
5.192741955 |
| Sample Variance |
122.004495 |
26.96456901 |
| Kurtosis |
-0.77667598 |
-0.352537913 |
| Skewness |
-0.086798138 |
0.144084526 |
| Range |
41 |
22 |
| Minimum |
50 |
75 |
| Maximum |
91 |
97 |
| Sum |
4327 |
5256 |
| Count |
62 |
62 |
Measurement scale.
The values for the two variables are quantitative in nature and have a point of origin i.e. zero. As such, they assume a ratio scale of measurement.
Measure of central tendency.
The measures of center include the mean, median, and the mode. The mean indicates the center of the data by revealing the most typical value in a group of data. The mean Group A Prior Training Scores is 69.790 while the average Group B Revised Training Scores is 84.77. The median indicates the center by revealing the middle most value in a group of data arranged in ascending order. The median Group A Prior Training Scores is 70 while the median Group B Revised Training Scores is 85. The mode shows the most frequent value. The modal Group A Prior Training Scores is 80 while the Group B Revised Training Scores is 85.
Evaluation.
The standard deviation shows the dispersion of data from the mean. The standard deviation for Group A Prior Training Scores is 11.04 while the standard deviation for the Group B Revised Training Scores is 5.19. The variance indicates the mean square deviations of the data points from the mean. The variance for Group A Prior Training Scores is 122 while the variance for Group B Revised Training Scores is 26.96 . The coefficient of skewness reveals the shape of the distribution of the data relative to the normal curve. The skewness for Group A Prior Training Scores is -0.09 implying that the data points are slightly negatively skewed while the skewness for Group B Revised Training Scores is 0.144 implying that the data points are slightly positively skewed.
The histogram for the dependent variable appears to approximately assume a normal distribution. Additionally, the frequency table indicates no presence of outliers. Lastly, the variables assume ratio scales of measurement. Assumptions for parametric statistical testing are met.
Dependent Samples (Paired-Samples) t Test: Descriptive Statistics and Assumption Testing
Frequency distribution table
|
Bin range |
Frequency |
|
10 |
3 |
|
20 |
6 |
|
30 |
9 |
|
40 |
13 |
|
50 |
17 |
|
60 |
1 |
| More |
0 |
Histogram.
Descriptive statistics table.
|
Pre-Exposure μg/dL |
Post-Exposure μg/dL |
|
| Mean |
32.85714286 |
33.28571429 |
| Standard Error |
1.752306546 |
1.781423416 |
| Median |
35 |
36 |
| Mode |
36 |
38 |
| Standard Deviation |
12.26614582 |
12.46996391 |
| Sample Variance |
150.4583333 |
155.5 |
| Kurtosis |
-0.576037127 |
-0.654212507 |
| Skewness |
-0.425109654 |
-0.483629097 |
| Range |
50 |
50 |
| Minimum |
6 |
6 |
| Maximum |
56 |
56 |
| Sum |
1610 |
1631 |
| Count |
49 |
49 |
Measurement scale. The values for the two variables are quantitative in nature and have a point of origin i.e. zero. As such, they assume a ratio scale of measurement.
Measure of central tendency.
The measures of center include the mean, median, and the mode. The mean indicates the center of the data by revealing the most typical value in a group of data. The mean Pre-Exposure μg/dL is 32.857 while the average Post-Exposure μg/dL is 33.286. The median indicates the center by revealing the middle most value in a group of data arranged in ascending order. The median Pre-Exposure μg/dL is 35 while the median Post-Exposure μg/dL is 36. The mode shows the most frequent value. The modal Pre-Exposure μg/dL is 36 while the Post-Exposure μg/dL is 38.
Evaluation. The standard deviation shows the dispersion of data from the mean. The standard deviation for Pre-Exposure μg/dL is 12.266 while the standard deviation for the Post-Exposure μg/dL is 12.47. The variance indicates the mean square deviations of the data points from the mean. The variance for Pre-Exposure μg/dL is 150.458 while the variance for Post-Exposure μg/dL is 155.5 . The coefficient of skewness reveals the shape of the distribution of the data relative to the normal curve. The skewness for Pre-Exposure μg/dL is -0.425 implying that the data points are negatively skewed while the skewness for Post-Exposure μg/dL is 0.144 implying that the data points are negatively skewed.
The histogram for the dependent variable appears to approximately assume a normal distribution. Additionally, the frequency table indicates no presence of outliers. Lastly, the variables assume ratio scales of measurement. Assumptions for parametric statistical testing are met.
ANOVA: Descriptive Statistics and Assumption Testing
Frequency distribution table
|
Bin range |
Frequency |
|
3 |
1 |
|
4 |
3 |
|
5 |
7 |
|
6 |
6 |
|
7 |
2 |
|
8 |
1 |
| More |
0 |
Histogram.
Descriptive statistics table.
|
A = Air |
B = Soil |
C = Water |
D = Training |
|
| Mean |
8.9 |
9.1 |
7 |
5.4 |
| Standard Error | 0.68 | 0.39 | 0.58 | 0.27 |
| Median |
9 |
9 |
6 |
5 |
| Mode |
11 |
8 |
6 |
5 |
| Standard Deviation | 3.06 | 1.74 | 2.58 | 1.19 |
| Sample Variance | 9.36 | 3.04 | 6.63 | 1.41 |
| Kurtosis | (0.63) | 0.12 | (0.24) | 0.25 |
| Skewness | (0.36) | 0.49 | 0.76 | 0.16 |
| Range |
11 |
7 |
9 |
5 |
| Minimum |
3 |
6 |
3 |
3 |
| Maximum |
14 |
13 |
12 |
8 |
| Sum |
178 |
182 |
140 |
108 |
| Count |
20 |
20 |
20 |
20 |
Measurement scale.
The data variables assume are quantitative in nature and assume a ratio scale of measurement.
Measure of central tendency.
The measures of center include the mean, median, and the mode. The mean indicates the center of the data by revealing the most typical value in a group of data. The mean ROI in air is 8.9%, in soil is 9.1%, in water is 7% and in training is 5.4%. The median indicates the center by revealing the middle most value in a group of data arranged in ascending order. The median ROI is Air, Soil, Water and training is 9%, 9%, 6%, and 5%, respectively. The mode shows the most frequent value. The mode for ROI is Air, Soil, Water and training is 11%, 8%, 6%, and 5%, respectively
Evaluation. The standard deviation shows the dispersion of data from the mean. The standard deviation for ROI is Air, Soil, Water and training is 3.06%, 1.74%, 2.58%, and 1.19%, respectively. The variance indicates the mean square deviations of the data points from the mean. The variance for ROI is Air, Soil, Water and training is 9.36%, 3.04%, 6.63%, and 1.41%, respectively. The coefficient of skewness reveals the shape of the distribution of the data relative to the normal curve. The skewness for ROI is Air, Soil, Water and training is -0.36%, 0.49%, 0.76%, and 0.16%, respectively. As such, the ROI for Air is negatively skewed, while the ROI for soil, water, and training is positively skewed.
The histogram for the dependent variable appears to approximately assume a normal distribution. Additionally, the frequency table indicates no presence of outliers. Lastly, the variables assume ratio scales of measurement. As such, assumptions for parametric statistical testing are met.
Data Analysis: Hypothesis Testing
Correlation: Hypothesis Testing
Restate the hypotheses:
Ho 1 : There is no statistically significant relationship between microns and mean annual sick days per employee.
Ha 1 : There is a statistically significant relationship between microns and mean annual sick days per employee.
Excel output
|
SUMMARY OUTPUT
|
||||||||||
|
Regression Statistics |
||||||||||
| Multiple R |
0.715984185 |
|||||||||
| R Square |
0.512633354 |
|||||||||
| Adjusted R Square |
0.507807941 |
|||||||||
| Standard Error |
1.327783455 |
|||||||||
| Observations |
103 |
|||||||||
| ANOVA | ||||||
|
df |
SS |
MS |
F |
Significance F |
||
| Regression |
1 |
187.2953239 |
187.2953 |
106.2362 |
1.89059E-17 |
|
| Residual |
101 |
178.0638994 |
1.763009 |
|||
| Total |
102 |
365.3592233 |
||||
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
|
| Intercept |
10.08144483 |
0.315156969 |
31.98865 |
0.000 |
9.456258184 |
10.70663 |
| microns |
-0.522376554 |
0.050681267 |
-10.3071 |
1.89E-17 |
-0.622914554 |
-0.42184 |
Notably, the Pearson correlation coefficient f r = -0.71598 indicating a strong positive correlation between microns and mean annual sick days per employee. This results in an r 2 of 0.5126 explaining 51.26% of the variations between the two variables.
Deploying a significance level of 0.05, the results show a p-value of 0.000 <0.05. There are sufficient grounds for rejecting the null hypothesis in favor of the alternative hypothesis. In this regard, there is a statistically significant relationship between microns and mean annual sick days per employee.
Simple Regression: Hypothesis Testing
Restate the hypotheses:
Ho 2 : The slope of the regression is equal to zero
Ha 2 : The slope of the regression is not equal to zero
Excel output
| SUMMARY OUTPUT | ||||||
|
Regression Statistics |
||||||
| Multiple R |
0.939559 |
|||||
| R Square |
0.882772 |
|||||
| Adjusted R Square |
0.882241 |
|||||
| Standard Error |
24.61329 |
|||||
| Observations |
223 |
|||||
| ANOVA | ||||||
|
df |
SS |
MS |
F |
Significance F |
||
| Regression |
1 |
1008202 |
1008202 |
1664.211 |
7.7E-105 |
|
| Residual |
221 |
133884.9 |
605.814 |
|||
| Total |
222 |
1142087 |
||||
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
|
| Intercept |
273.4494 |
2.665262 |
102.5976 |
2.1E-188 |
268.1968 |
278.702 |
| safety training expenditure |
-0.14337 |
0.003514 |
-40.7947 |
7.7E-105 |
-0.15029 |
-0.13644 |
The multiple r which is equivalent to the Pearson coefficient of correlation is 0.939 indicating that the relationship between the two variables is very strong. This gives an r-squared of 0.8828 indicating that 88.28% of the variations of the dependent variables is explained by the regression model ( Draper & Smith, 2014) .
Deploying an alpha of 0.05, the ANOVA F value is 1664.21 and a p-value of 0.000 < 0.05. This indicates strong evidence for rejecting the null hypothesis. The slope of regression is significantly different from zero. The regression equation is given by; Y =-0.1434X + 273.449. So for every unit increase in the lost time hours, the safety training expenditure decreases by 0.1434. The p-value of the regression coefficient is 7,7E-105 < 0.05 level of significance. The regression coefficient is statistically significant.
Multiple Regression: Hypothesis Testing
Restate the hypotheses:
H0 3 : There is no statistically significant relationship between the X variables and the Y variable
Ha 3 : There is a statistically significant relationship between the X variables and the Y variable
| SUMMARY OUTPUT | ||||||
|
Regression Statistics |
||||||
| Multiple R |
0.601842 |
|||||
| R Square |
0.362214 |
|||||
| Adjusted R Square |
0.360083 |
|||||
| Standard Error |
5.518566 |
|||||
| Observations |
1503 |
|||||
| ANOVA | ||||||
|
df |
SS |
MS |
F |
Significance F |
||
| Regression |
5 |
25891.89 |
5178.378 |
170.0361 |
2.1E-143 |
|
| Residual |
1497 |
45590.49 |
30.45457 |
|||
| Total |
1502 |
71482.38 |
||||
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
|
| Intercept |
126.8225 |
0.62382 |
203.2997 |
0 |
125.5988 |
128.0461 |
| Frequency (Hz) |
-0.00112 |
4.76E-05 |
-23.4885 |
4.1E-104 |
-0.00121 |
-0.00102 |
| Angle in Degrees |
0.047342 |
0.037308 |
1.268957 |
0.204654 |
-0.02584 |
0.120524 |
| Chord Length |
-5.49532 |
2.927962 |
-1.87684 |
0.060734 |
-11.2387 |
0.248026 |
| Velocity (Meters per Second) |
0.08324 |
0.0093 |
8.950317 |
1.02E-18 |
0.064997 |
0.101482 |
| Displacement |
-240.506 |
16.51903 |
-14.5593 |
5.21E-45 |
-272.909 |
-208.103 |
Notably, the multiple r is 0.6018 indicating a moderately strong relationship among the variables. The R-squared value is 0.3622 indicating that the multiple regression model explains 36.22% of the variation of the dependent variable around its mean.
Employing an alpha of 0.05, the ANOVA F value is 170.04 and a P-value of 0.000 < 0.05. The null hypothesis is rejected in favor of the alternative hypothesis. This implies that there is a significant relationship between the X variables and the Y variable. That is, the Y values that would be predicted by the regression model are closer to the actual values that would be expected to be obtained by chance. All the coefficients of independent variables except Frequency (Hz) are statistically significant as shown by p-values less than 0.05 level of significance. The coefficient for Frequency (H) has a P-value of 0.205 > 0.05 indicating that the coefficient is not statistically significant.
The regression model is given by;
Y =126.82 -240.506*(Displacement)+0.083*(Velocity)-5.495*(Chord length)+0.0473*(Angle degrees)-0.0011*Frequency
Hypothesis testing looks for significant relationships between variables or significant differences between variables or groups. The t-Test is used to compare two means and is the simplest form of a test of differences. While, ANOVA is used to compare more than two means. Below are some samples of both t-test and ANOVA hypothesis testing.
Independent Samples t -Test: Hypothesis Testing
H0: There is no statistically significant difference in the mean values for the training scores between Group A (Prior) and Group B (Revised).
Ha: There is a statistically significant difference in the mean values for the training scores between Group A (Prior) and Group B (Revised).
Excel output
| t-Test: Two-Sample Assuming Unequal Variances | ||
Group A Prior Training Scores |
Group B Revised Training Scores |
|
| Mean |
69.79032258 |
84.77419355 |
| Variance |
122.004495 |
26.96456901 |
| Observations |
62 |
62 |
| Hypothesized Mean Difference |
0 |
|
| Df |
87 |
|
| t Stat |
-9.666557191 |
|
| P(T<=t) one-tail |
0.000000 |
|
| t Critical one-tail |
1.662557349 |
|
| P(T<=t) two-tail |
0.00000 |
|
| t Critical two-tail |
1.987608282 |
|
Notably, the results indicate that the mean for training scores for Group A is lower compared to the mean for the training scores for Group B. The P-value for the test is 0.000 < 0.05. This indicates that the null hypothesis is rejected and as such, there is a statistically significant difference in the mean values for the training scores between Group A (Prior) and Group B (Revised) ( Cohen, 2013) .
Dependent Samples (Paired Samples) t -Test: Hypothesis Testing
H0 : There is no statistically significant difference in the mean values between Pre and Post-Exposure
Ha: There is a statistically significant difference in the mean values between Pre and Post-Exposure
| t-Test: Paired Two Sample for Means | ||
|
Pre-Exposure μg/dL |
Post-Exposure μg/dL |
|
| Mean |
32.85714 |
33.28571 |
| Variance |
150.4583 |
155.5 |
| Observations |
49 |
49 |
| Pearson Correlation |
0.992236 |
|
| Hypothesized Mean Difference |
0 |
|
| df |
48 |
|
| t Stat |
-1.9298 |
|
| P(T<=t) one-tail |
0.029776 |
|
| t Critical one-tail |
1.677224 |
|
| P(T<=t) two-tail |
0.059553 |
|
| t Critical two-tail |
2.010635 |
|
Observably, the employee’s mean value for Post-exposure is greater compared to Pre-Exposure. However, the P-value is 0.0595 > 0.05 implying that the null hypothesis is not rejected ( Cohen, 2013) . As such, there is no significant difference in the employee Pre-exposure and Post-exposure μg/dL scores.
ANOVA: Hypothesis Testing
H0: The mean investment return is the same for all groups
Ha: The mean return on investment is not the same for all groups
Excel output
| ANOVA: Single Factor | ||||||
| SUMMARY | ||||||
|
Groups |
Count |
Sum |
Average |
Variance |
||
| A = Air |
20 |
178 |
8.9 |
9.357895 |
||
| B = Soil |
20 |
182 |
9.1 |
3.042105 |
||
| C = Water |
20 |
140 |
7 |
6.631579 |
||
| D = Training |
20 |
108 |
5.4 |
1.410526 |
||
| ANOVA | ||||||
Source of Variation |
SS |
df |
MS |
F |
P-value |
F crit |
| Between Groups |
182.8 |
3 |
60.93333 |
11.9231 |
0.0000 |
2.724944 |
| Within Groups |
388.4 |
76 |
5.110526 |
|||
| Total |
571.2 |
79 |
||||
Notably, the mean return on investment for training consulting is the lowest. However, the p-value for ANOVA is 0.000 <0.05 implying that the null hypothesis is rejected. The mean return on investment is not the same for all consulting projects.
References
Cohen, J. (2013). Statistical power analysis for the behavioral sciences . Routledge.
Mendenhall, W. M., Sincich, T. L., & Boudreau, N. S. (2016). Statistics for Engineering and the Sciences, Student Solutions Manual . Chapman and Hall/CRC.
Mooi, E., Sarstedt, M., & Mooi-Reci, I. (2018). Descriptive Statistics. In Market Research (pp. 95-152). Springer, Singa
Draper, N. R., & Smith, H. (2014). Applied regression analysis (Vol. 326). John Wiley & Sons.
