Question

An independent random sampling design was used to compare the means of six treatments based on samples of four observations per treatment. The pooled estimator of \(\sigma^{2}\) is \(9.12,\) and the sample means follow: $$ \begin{array}{lll} \bar{x}_{1}=101.6 & \bar{x}_{2}=98.4 & \bar{x}_{3}=112.3 \\ \bar{x}_{4}=92.9 & \bar{x}_{5}=104.2 & \bar{x}_{6}=113.8 \end{array} $$ a. Give the value of \(\omega\) that you would use to make pairwise comparisons of the treatment means for \(\alpha=.05\) b. Rank the treatment means using pairwise comparisons.

Short answer

Answer: The value of omega used to make pairwise comparisons of means for six treatments at α = 0.05 is 6.15. Based on these pairwise comparisons, the treatment means are ranked in the following order (from highest to lowest): 6, 3, 5, 1, 2, and 4.

Step-by-step solution

(Step 1: Understand the problem)

We need to make pairwise comparisons of the means for six treatments and then rank them. For this, we will use the Tukey's Honest Significant Difference (HSD) test.

(Step 2: Compute the parameter values needed for HSD test)

To perform Tukey's HSD test, we need to calculate the pooled variance and the degrees of freedom. We are given the pooled estimator of \(\sigma^2\) as \(9.12\). The number of treatments is 6, and the number of observations per treatment is 4, so the total number of observations is \(6 \times 4 = 24\). The degrees of freedom (df_total) is total number of observations minus the number of treatments: df_total = \(24-6 = 18\).

(Step 3: Calculate q-value)

To find the \(\omega\) value, we need to calculate the q-value from the Studentized range distribution table. For \(\alpha = 0.05\), the critical value of q with 6 treatments and 18 degrees of freedom is around \(4.038\) (can be found in q-tables).

(Step 4: Calculate the HSD value)

To perform the Tukey's HSD test and find \(\omega\) value, we use the following formula: \(\omega = q \cdot \sqrt{\frac{\sigma^2}{n}}\). Plugging in the values, we get: \(\omega = 4.038 \times \sqrt{\frac{9.12}{4}} = 6.15\).

(Step 5: Rank the treatment means using pairwise comparisons)

Pairwise comparisons can be made using the HSD value (\(\omega = 6.15\)). If the absolute difference between two treatment means is greater than or equal to \(\omega\), it is considered significantly different. In this case, since 6.15 is a maximum difference beyond which the treatment means are considered significantly different, we rank the means as follows (from highest to lowest): 1. \(\bar{x}_{6}=113.8\) 2. \(\bar{x}_{3}=112.3\) 3. \(\bar{x}_{5}=104.2\) 4. \(\bar{x}_{1}=101.6\) 5. \(\bar{x}_{2}=98.4\) 6. \(\bar{x}_{4}=92.9\) So, the ranking of the treatment means is 6, 3, 5, 1, 2, and 4.

Key concepts

Tukey's Honest Significant Difference (HSD)

Tukey's Honest Significant Difference (HSD) is a statistical test used to determine if there are significant differences between the means of different treatments or groups. This method is particularly useful when you've conducted an ANOVA test and found that at least one group mean is different from the others. But ANOVA doesn't tell you which specific means are different. Here's where Tukey's HSD steps in to help.

The HSD test uses a specific formula to calculate a critical value or threshold known as the HSD value. If the difference between any two sample means exceeds this value, the difference is deemed statistically significant. In simpler terms, it tells us whether the differences we see between group means are genuine or if they're just due to random chance.

• Helps identify specific group differences.
• Utilizes ANOVA as a prerequisite stepping stone.
• Provides a clear threshold for significance.

Understanding this fundamental concept allows researchers to delve deeper into their data analysis and make informed decisions about their findings.

Pairwise Comparison

Pairwise comparison involves comparing every pair of group means to decide whether there is a significant difference between them. In the context of Tukey's HSD test, pairwise comparisons form the core of the analysis. Essentially, you will take each treatment mean and compare it to every other treatment mean, one pair at a time.

For example, if you have six treatment means, you will be making comparisons like: Mean 1 vs. Mean 2, Mean 1 vs. Mean 3, and so on until all possible unique pairs are compared.

• Enhances detailed insight into group differences.
• Relies on a pre-defined significance threshold (HSD value).
• Provides a structured approach to understanding mean differences.

By making these pairwise comparisons, you get a comprehensive picture of how treatment means relate to one another in terms of statistical significance.

Treatment Means

Treatment means are the average values of a specific group or category in a set of experiments. Each mean represents the central tendency of the data for that treatment group. In the given exercise, there are six treatment means, each representing a different group. These are the values that you analyze using pairwise comparisons to see how they differ from one another.

In statistical analysis, especially when using the HSD test, it's crucial to understand that treatment means are the primary elements under review. Their comparison can reveal differences in performance or response under varying conditions.

• Represent average results from different groups.
• Central to testing and comparison in ANOVA.
• Provide insights into how treatments vary in effect.

Understanding treatment means is essential to grasp their implications and significance in the broader statistical analysis, such as the Tukey's test.

Statistical Significance

Statistical significance is a critical concept in hypothesis testing, indicating whether the observed effect in your data is genuine or occurred by random chance. When conducting an ANOVA followed by a Tukey's HSD test, researchers are essentially seeking statistical significance to validate differences between group means.

In simpler terms, if the result is statistically significant, it means there's a high probability that the result reflects a true effect rather than random noise. The threshold for significance is usually set before the analysis begins, commonly at \(\alpha = 0.05\) or 5%. If your pairwise comparison exceeds the Tukey's HSD critical value (or \(\omega\) in our exercise), it achieves statistical significance.

• Measures the reliability of test results.
• Dictates whether differences are substantial.
• Uses predefined criteria (e.g., \(\alpha = 0.05\)).

Recognizing statistical significance helps researchers make confident conclusions about their research findings.