A Consumer Organization Wants To Know If There Is A Differen
1 A Consumer Organization Wants To Know If There Is A Difference In T
A consumer organization aims to determine whether there are significant differences in the prices of a particular toy across three different types of stores: discount toy stores, variety stores, and department stores. To investigate this, samples of five stores from each category were examined, and the prices collected were analyzed using an Analysis of Variance (ANOVA). The data provided include the mean prices, standard deviations, and sample sizes for each store type, along with the results from the ANOVA table.
The specific data are as follows:
- Discount Toy Stores: Mean = 13.3, Standard Deviation = 6.4, Sample Size = 5
- Variety Stores: Mean = 16.4, Standard Deviation = 6.4, Sample Size = 5
- Department Stores: Mean = 18.6, Standard Deviation = 6.4, Sample Size = 5
The ANOVA table provides the sum of squares (SS), degrees of freedom (df), mean square (MS), the F-statistic, and the p-value for the treatment effect, which tests for differences among the store means.
Analysis and Interpretation
Statistically, the null hypothesis in ANOVA is that all group means are equal, while the alternative hypothesis is that at least one group mean differs. The key question is whether the observed variations truly reflect differences in store prices or could have arisen by random chance.
According to the ANOVA results, the F-statistic is calculated from the mean squares of the treatment (between-group variability) and the error (within-group variability). Specifically, the treatment mean square (MS) is given as 0.009, and the error mean square (MS) is 28.367, although the exact F-statistic value is not fully displayed; it can be derived from the MS values. The p-value associated with this F-statistic indicates the probability that the observed differences or more extreme differences could occur if the null hypothesis were true.
Is there a significant difference at the 0.05 level?
Since the p-value is not explicitly provided, but the significance level is set at 0.05, decision-making revolves around the comparison of the p-value to 0.05. If the p-value is less than 0.05, we reject the null hypothesis, concluding that there are significant differences in average toy prices among the three types of stores. Conversely, if the p-value exceeds 0.05, there is insufficient evidence to claim that the store types differ significantly in pricing.
Why was ANOVA used?
ANOVA was employed because the research involves comparing the means of more than two groups simultaneously. Unlike multiple t-tests, which increase the risk of Type I error when performing multiple comparisons, ANOVA efficiently tests whether at least one group mean differs from the others while controlling the overall error rate. Furthermore, ANOVA handles the variability within and between groups in a structured manner, providing a robust framework for analyzing these differences.
Conclusion
Based on the data and the results provided, if the p-value derived from the ANOVA table is less than 0.05, it indicates that there is a statistically significant difference amongst the store types regarding toy prices. If not, the prices are considered statistically similar across the store categories. Given the summarized data, the initial conclusion suggests that store type may influence the price, but the definitive decision depends on the p-value which typically would be included in the detailed ANOVA output.
References
- Montgomery, D. C. (2017). Design and Analysis of Experiments. John Wiley & Sons.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage.
- Anderson, D. R., Sweeney, D. J., & Williams, T. A. (2016). Statistics for Business and Economics. Cengage Learning.
- Gravetter, F. J., & Wallnau, L. B. (2016). Statistics for the Behavioral Sciences. Cengage Learning.
- Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson.