# Assignment: One-Way ANOVA Essay

## Assignment: One-Way ANOVA Essay

**Assignment: One-Way ANOVA Essay**

Statistical tests are an essential tool used to help understand the meaning of a given dataset. Based on the statistical tests results, one can deduce whether the data obtained from a study supports the hypothesis or not. Similarly, they help in determining the credibility and inferential ability of the data collected from the sample populations in a study. Various tests serve different purposes and can only be applied depending on the nature of the data involved and the research questions to be answered. ANOVA test is commonly used to determine the correlation between given variables. The data from the grades.sav.set will be used in this study to explore further the use of ANOVA in data analysis and interpretation.

#### Section 1: Data File Description

The data in this study were obtained from the SPSS dataset. A teacher recorded the performance and demographic characteristics of a group of students in the class. A total of 105 students participated in the study. The variables analyzed included the section and quiz3 outcomes which were displayed in the SPSS data. The participants were divided into three groups labeled 1, 2 and 3 with each section having 33, 39 and 33 students respectively. The section variables were used as predictor variables and were measured in nominal scale. On the other hand, quiz3 variable was the outcome variable measured in ratio scale.

#### Section 2: Testing Assumptions

ANOVA tests are only suitable under given assumptions which must be considered when analyzed data. The first assumption for this case was that the data were independent and followed a normal distribution. Before the ANOVA test is performed, it is important to identify an assumption that may come along with the test. Secondly, it was also assumed that the data are centrally located and does not have outliers.

**Figure 1.1: **SPSS histogram of quiz3.

** **Histogram was constructed from the collected data and this would help determine whether the data followed a normal

distribution or not. Furthermore, it could also be used in testing the second assumption, which was on the tendency of central location. The figure above shows that the total sample population (N) was 105 with the mean and standard deviation of 7.13 and 1.16 respectively. The findings confirm that the data follows a normal distribution and there are no outliers considering that the tails are evenly sloped. In addition, the descriptive statistics were also used in testing the study assumptions. The results indicated that the data were negatively skewed while the kurtosis was positive. However, the kurtosis value is not high and this means that there is nearly equal distribution at both the tail and the head.

Descriptive Statistics |
|||||

N | Skewness | Kurtosis | |||

Statistic | Statistic | Std. Error | Statistic | Std. Error | |

quiz3 | 105 | -.078 | .236 | .149 | .467 |

Valid N (listwise) | 105 |

**Figure 1.2: **SPSS output of quiz3 descriptive statistics with skewness and kurtosis

As evident in the diagram above, the kurtosis and skewness values are within the acceptable limits and thus appropriate.

Tests of Normality |
||||||

Kolmogorov-Smirnov^{a} |
Shapiro-Wilk | |||||

Statistic | df | Sig. | Statistic | df | Sig. | |

quiz3 | .143 | 105 | .000 | .948 | 105 | .000 |

a. Lilliefors Significance Correction |

**Figure 1.3: **SPSS output of the Shapiro-Wilk test results for quiz3

The test for normality was also performed and the p-value was 0.000 which is significant at *p* < .05. The p-value obtained is significant and thus the null hypothesis is rejected.

Test of Homogeneity of Variances |
|||

quiz3 | |||

Levene Statistic | df1 | df2 | Sig. |

2.898 | 2 | 102 | .060 |

**Figure 1.4: **SPSS output results of the Levene test

The Levene test was used to determine the equality of variance between the variables. The results as indicated in fig 1.4 show that there is no significant violation of the homogeneity of variance considering that the Sig. is at a .060 and this is greater than .05.

**Assumptions**

** **All the assumption outlined in the study were met and thus, the tests were reliable. The reliability of the tests used in data analysis also influences the usefulness of the final results; in this case, the normality test, Shapiro-Wilk test, Levene tests, and the histogram show normal distribution and evenness of the variance between variables.

#### Section 3: Research Question, Hypotheses, and Alpha Level

The research questions for the study included: is there a significant variation between the mean of sore for the quiz3 among the three groups? With the research question, the formulated alternative hypothesis was that there is a significant difference in the mean score for the three groups (sections). The null hypothesis was that there is no significant difference in the mean score for the three sections. The confidence interval used in this case was 95% with the alpha critical value being 0.05.

#### Section 4: Interpretation

**Figure 1.5: **SPSS output data mean of quiz3 for each section

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The above figure (fig 1.5) shows that there is a significant difference in the quiz3 score for the three sections. The highest score was recorded by section 3, followed by 1 and then 2.

Descriptives |
||||||||

quiz3 | ||||||||

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

Lower Bound | Upper Bound | |||||||

1 | 33 | 7.27 | 1.153 | .201 | 6.86 | 7.68 | 5 | 10 |

2 | 39 | 6.33 | 1.611 | .258 | 5.81 | 6.86 | 2 | 10 |

3 | 33 | 7.94 | 1.560 | .272 | 7.39 | 8.49 | 6 | 10 |

Total | 105 | 7.13 | 1.600 | .156 | 6.82 | 7.44 | 2 | 10 |

**Figure 1.6: **SPSS output data of descriptives for quiz3 including all sections.

From the descriptive statistics above, the mean value for section 1,2 and 3 were 7.27, 6.33 and 7.94 respectively. The standard deviation value also varied from one section to another; the values obtained for section 1, 2 and 3 were 1.153, 1.611 and 1.1560 respectively. The results confirm that there is a significant variation in the quiz score among the three sections.

ANOVA |
|||||

quiz3 | |||||

Sum of Squares | df | Mean Square | F | Sig. | |

Between Groups | 47.042 | 2 | 23.521 | 10.951 | .000 |

Within Groups | 219.091 | 102 | 2.148 | ||

Total | 266.133 | 104 |

**Figure 1.7:** SPSS output data showing ANOVA results

The ANOVA results as depicted in fig 1.7 shows that the degree of freedom for the between and within groups is 2 and 102 respectively. The total degree of freedom value is 104. The F score is 10.951 while the sig. value is .000 with the effect size of 0.5588 which is significantly small. The sig. value is 0.000 and this shows that the p-value is less than 0.05 and thus significant at the critical alpha value; therefore, the null hypothesis is rejected (George, and Mallery, 2016).

Multiple Comparisons |
||||||

Dependent Variable: quiz3 | ||||||

Tukey HSD | ||||||

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

Lower Bound | Upper Bound | |||||

1 | 2 | .939^{*} |
.347 | .021 | .11 | 1.76 |

3 | -.667 | .361 | .159 | -1.52 | .19 | |

2 | 1 | -.939^{*} |
.347 | .021 | -1.76 | -.11 |

3 | -1.606^{*} |
.347 | .000 | -2.43 | -.78 | |

3 | 1 | .667 | .361 | .159 | -.19 | 1.52 |

2 | 1.606^{*} |
.347 | .000 | .78 | 2.43 | |

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

**Figure 1.8: **SPSS output data showing post-hoc

Furthermore, the mean comparison between the three sections was one and the post-hoc used in determining whether there is a significant difference between the groups. From the results in fig 1.8, there is a significant variation between the scores in the three sections. The variance value for section 1 compared to section 2 is 0.939 which is quite significant. Similarly, the variance value for section 2 compared to section 3 was 1.606 which is still high. The least variance value was obtained between section 1 and 3.

#### Section 5: Conclusion

Finally, the ANOVA results in this study indicated that there was a significant difference in the score between the three sections. The findings from the statistical test are helpful in testing the hypotheses and thus answering the research question. In this case, the null hypothesis was rejected because the p-value obtained was significant at the critical value alpha=0.05. The performance in quiz3 varied significantly from one group to another and based on this, a further study may be conducted to explore the factors contributing to the variation in the performances.

On the other hand, in as much as the test was appropriate and helpful in answering the research questions, it was faced by limitations such as small sample size. The sample size was not adequate to fully provide an answer to the research question. Furthermore, due to the differences within the data obtained, one is required to conduct separate tests such as post-hoc to discover the significance of the data when performing an ANOVA test; this appears to be cumbersome.

As with your previous assignments, you will complete this assignment with the DAA Template. Links to additional resources are available in the Resources area.

Reminder: The format of this SPSS assignment should be narrative with supporting statistical output (table and graphs) integrated into the text in the appropriate places (not all at the end of the document).

You will analyze the following variables in the grades.sav data set:

section

quiz3

Step 1: Write Section 1 of the DAA.

Provide the context of the grades.sav data set.

Include a definition of the specified variables (predictor, outcome) and corresponding scales of measurement.

Specify the sample size of the data set.

Step 2: Write Section 2 of the DAA.

Analyze the assumptions of the one-way ANOVA.

Paste the SPSS histogram output for quiz3 and discuss your visual interpretations.

Paste SPSS descriptives output showing skewness and kurtosis values for quiz3 and interpret them.

Paste SPSS output for the Shapiro-Wilk test of quiz3 and interpret it.

Report the results of the Levene test and interpret it.

Summarize whether or not the assumptions of the one-way ANOVA are met.

Step 3: Write Section 3 of the DAA.

Specify a research question related to the one-way ANOVA.

Articulate the null hypothesis and alternative hypothesis.

Specify the alpha level.

Step 4: Write Section 4 of the DAA.

Begin by pasting SPSS output of the means plot and providing an interpretation.

Also report the means and standard deviations of quiz3 for each level of the section variable.

Next, paste the SPSS ANOVA output and report the results of the F test, including:

Degrees of freedom.

F value.

p value.

Calculated effect size.

Interpretation of the effect size.

Finally, if the omnibus F is significant, provide the SPSS post-hoc (Tukey HSD) output.

Interpret the post-hoc tests.

Step 5: Write Section 5 of the DAA.

Discuss the conclusions of the one-way ANOVA as it relates to the research question.

Conclude with an analysis of the strengths and limitations of one-way ANOVA.

Submit your DAA Template as an attached Word document in the assignment area.

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