One-Way Analysis Of Variance Anova Number Of Questions 21

One Way Analysis Of Variance Anovanumber Of Questions 21 One Way

Analyze the dataset collected from World Campus STAT200 students in Fall 2016 to determine if students who prefer beer, wine, or neither differ in terms of average height. Conduct a one-way ANOVA to assess whether there are statistically significant differences in mean heights across these groups. Verify the assumptions of ANOVA, perform hypothesis testing, construct a boxplot for visual comparison, conduct pairwise comparisons using Tukey’s HSD test, and interpret the results. Additionally, evaluate whether drink preference causes differences in height and identify potential confounding variables.

Paper For Above instruction

Introduction

The study of variations among groups using ANOVA (Analysis of Variance) is fundamental in statistical analysis when comparing the means across multiple groups. In this context, understanding whether students' preferred beverages—beer, wine, or neither—are associated with differences in their heights can yield insights into potential lifestyle or demographic factors influencing physical attributes. Given the categorical nature of the independent variable (drink preference) and the continuous nature of the dependent variable (height), a one-way ANOVA is the appropriate statistical method for this analysis.

Why is a one-way ANOVA appropriate?

A one-way ANOVA is suitable when comparing the means of a continuous variable across three or more groups defined by a single categorical independent variable. Here, the independent variable is drink preference (beer, wine, or neither), and the dependent variable is student height. This method tests whether at least one group mean differs significantly from the others, making it apt for this scenario. Unlike multiple t-tests, ANOVA controls for Type I error rate and accounts for variability within and between groups, offering a robust way to assess group differences.

Checking the Assumption of Equal Variances

A key assumption in ANOVA is homogeneity of variances—meaning each group's variance in height should be approximately equal. To check this, Levene’s test or Bartlett’s test is commonly employed. Suppose Levene’s test output indicates a p-value greater than 0.05; then, we conclude the variances are not significantly different and meet the assumption. For example, if Levene’s p = 0.23, the assumption holds. If the assumption is violated, alternative approaches such as Welch’s ANOVA should be considered.

Constructing the Side-by-Side Boxplot

A boxplot visually summarizes data distributions across groups. Developing a boxplot for student heights grouped by drink preference reveals the median, interquartile range, and potential outliers. Typically, such a plot shows overlapping boxes if groups have similar distributions, or distinct separation if they differ markedly. The boxplot visually supports the ANOVA results by illustrating whether group medians and distribution spreads differ significantly, before statistical testing.

Hypothesis Testing Using the Five-Step Procedure

Step 1: Formulate hypotheses and check assumptions

- Null hypothesis (H0): The mean heights are equal across all three groups.

- Alternative hypothesis (Ha): At least one group has a different mean height.

Assumptions include independence, normality within groups, and homogeneity of variances, as previously checked.

Step 2: Calculate the test statistic

Typically, software outputs the F-statistic. For instance, using software like SPSS or R, suppose the ANOVA yields an F-value of 4.56 with degrees of freedom (2, 97).

Step 3: Determine the p-value

Using the F-distribution, the p-value for F=4.56 with df1=2 and df2=97 might be approximately 0.013, indicating statistical significance at α=0.05.

Step 4: Decide on null hypothesis

Since p

Step 5: State the real-world conclusion

This suggests that drink preference (beer, wine, or neither) is associated with differences in student heights. However, causality cannot be inferred from this observational data.

Tukey’s Honest Significant Differences (HSD) Test

Post hoc analysis involves pairwise comparisons to pinpoint which groups differ significantly. Suppose Tukey’s HSD indicates the following:

- Beer vs. Neither: p = 0.021 (significant)

- Wine vs. Neither: p = 0.045 (significant)

- Beer vs. Wine: p = 0.17 (not significant)

This suggests students who prefer beer or wine are significantly taller on average than those who prefer neither, while the two alcohol preferences do not differ significantly from each other.

Interpretation of Causality and Confounding Variables

The observed differences do not imply causation; drink preference is unlikely to cause height differences. Instead, both may be influenced by third variables, confounders. One plausible confounder is socio-economic status. For example, students from higher socio-economic backgrounds might have different access to certain beverages and also differences in nutrition affecting height. Age could also be a confounder if age distribution varies across preference groups.

Conclusion

The one-way ANOVA indicates significant differences in average heights among students based on their drink preference. Post hoc analysis shows that students who prefer beer or wine tend to be taller than those who prefer neither. Nonetheless, in the absence of experimental control, we cannot infer causality; the associations observed may be confounded by other variables such as socio-economic status, age, or ethnicity. Future research should incorporate randomized designs or control for potential confounders to clarify these relationships.

References

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