Question

A botanist used a completely randomized design to allocate 45 individually potted eggplant plants to five different soil treatments. The observed variable was the total plant dry weight without roots \((\mathrm{gm})\) after 31 days of growth. The treatment means were as shown in the following table. \({ }^{35}\) The MS(within) was \(0.2246 .\) Use Fisher's LSD intervals to compare all pairs of means at \(\alpha_{c_{\mathbb{Y}}}=0.05 .\) Present your results in a summary table similar to Table 11.9.4. (Hint: Take note that all sample sizes are equal; thus the calculated margin of error need only be calculated once for all comparisons. There are a total of 10 comparisons possible.) $$ \begin{array}{|llllll|} \hline \text { Treatment } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Mean } & 4.37 & 4.76 & 3.70 & 5.41 & 5.38 \\ n & 9 & 9 & 9 & 9 & 9 \\ \hline \end{array} $$

Short answer

The margin of error for the LSD test is calculated using the critical t-value and MS_{within}. The ME is then used to determine if the difference between each pair of treatment means is significant at the \(\alpha = 0.05\) level.

Step-by-step solution

Determine the Margin of Error

First, calculate the margin of error (ME) for the Fisher's LSD test using the formula: ME = t * sqrt(2 * MS_{within}/n), where t is the t-value from the t-distribution for the chosen alpha level and degrees of freedom (df = N - k), MS_{within} is the mean square within treatments, and n is the sample size for each treatment. Since sample sizes are equal and all comparisons will use the same ME, calculate this value only once.

Find the Critical t-value

Determine the critical t-value using a t-distribution table or software for \(\alpha = 0.05\) and \(df = 45 - 5 = 40\), since there are 45 observations and 5 treatments. For \(\alpha = 0.05\), two-tailed, the critical t-value is approximately 2.021.

Calculate the Margin of Error

Using the critical t-value and the MS_{within}, calculate the margin of error: ME = 2.021 * sqrt(2 * 0.2246 / 9).

Calculate LSD Intervals for All Pairs

For each pair of treatment means (A, B, C, D, E), calculate the Least Significant Difference (LSD) interval by subtracting and adding the ME to the difference of the pair of means. As there are 5 treatments, we have \(5\choose 2\) = 10 pairs to compare.

Summarize the Results

Present the differences between the means and their corresponding LSD intervals in a summary table. The entries should show whether the differences are significant or not; if the interval contains zero, the difference is not significant.

Key concepts

Completely Randomized Design

In the field of statistics, a completely randomized design is a fundamental experimental setup used to evaluate the effects of different treatments. It's preferred because of its simplicity and its ability to control for extraneous variables that could confound the results. Imagine you're a botanist wanting to test the effectiveness of five soil treatments on the growth of eggplant plants. By randomly assigning the plants to each treatment group, you ensure that each plant has an equal chance of receiving any given treatment, minimizing the risk of bias in your results.

It's like entering all candidates in a race and letting chance decide who starts in which lane. This random distribution of subjects, or in this case, plants, across groups means that variations within treatment results are more likely due to actual differences in treatments rather than other external factors, laying the foundation for an unbiased statistical analysis. It's crucial to note that in a completely randomized design, it is assumed that the experimental units are homogenous or similar enough that any variation among them will not significantly impact the outcome.

Mean Square Within (MS_within)

The term mean square within (often abbreviated to MS_within) is pivotal to analyzing variance within groups of a study. It's part of what's called an ANOVA table, which stands for Analysis of Variance. MS_within is essentially the average of the squared differences from the mean within each treatment group. It gives an estimate of the trial-to-trial variability and is indispensable in calculating the margin of error for the Fisher's LSD test.

In our botanical study, MS_within measures how much the dry weights of eggplant plants vary within the same soil treatment. Imagine MS_within as a gauge of the noise level in a room—higher MS_within means more noise and more variability in plant growth due to random fluctuations rather than the soil treatment.

t-Distribution

The t-distribution, also known as Student's t-distribution, is a probability distribution that estimatorss how an estimate of the mean of a normally distributed population is likely to behave when the size of the sample is small and the population variance is unknown. It's a foundational concept in small sample statistical hypotheses testing.

In practice, when comparing treatment means like in our plant study, the t-distribution is used to understand the range in which we expect the real mean to fall, given our sample data. It helps determine the critical value (t-value) which dictates the threshold for a statistically significant difference between treatment groups. It's tailored to the degrees of freedom in your analysis, which, put simply, are the number of values in the final calculation of a statistic that are free to vary.

Least Significant Difference (LSD)

The least significant difference or LSD is a post hoc test that tells us whether the difference between group means is statistically significant. It's a method of dissecting the data following an ANOVA when the null hypothesis has been rejected. To perform an LSD test, you calculate a value using the mean square within (MS_within), the t-distribution, and the sample size. This value acts as a yardstick, setting the minimum difference needed between two means to declare them significantly different from each other.

Applying LSD in our example involves calculating a margin of error and comparing every possible pair of soil treatments to see which ones actually affect the dry weight of our plants. When the difference between the means exceeds this calculated LSD value, the treatments are considered different enough to be significant, overcoming the inherent variability in the plants' responses to the treatments. However, it's important to exercise caution with LSD due to the risk of Type I error, especially with multiple comparisons.

Treatment Means Comparison

Comparing treatment means is the heart of many experiments, designed to identify whether the application of different treatments leads to changes in the outcome. In the case of our botanical study, the treatment means comparison involves assessing the dry weights of eggplant plants under different soil conditions to discover if there are any significant changes caused by the treatments.

By using statistical tests like Fisher's LSD, which considers mean, variability, and sample size, we can compare pairs of treatment groups. This direct comparison is crucial to conclude precisely which treatments produce different results and are suitable for practical applications. It's somewhat like a tournament where each contestant (treatment mean) faces off against its competitors to see who truly stands out, with the difference being that in our study, the true winners are the soil treatments that significantly contribute to plant growth.