Question

In an article in the Annals of Botany, a researcher reported the basal stem diameters of two groups of dicot sunflowers: those that were left to sway freely in the wind and those that were artificially supported. \({ }^{18}\) A similar experiment was conducted for monocot maize plants. Although the authors measured other variables in a more complicated experimental design, assume that each group consisted of 64 plants (a total of 128 sunflower and 128 maize plants). The values shown in the table are the sample means plus or minus the standard error. $$\begin{array}{l|c|c} & \text { Sunflower } & \text { Maize } \\\\\hline \text { Free-Standing } & 35.3 \pm .72 & 16.2 \pm .41 \\\\\text {Supported } & 32.1 \pm .72 & 14.6 \pm .40\end{array}$$ Use your knowledge of statistical estimation to compare the free-standing and supported basal diameters for the two plants. Write a paragraph describing your conclusions, making sure to include a measure of the accuracy of your inference.

Short answer

Question: Compare the basal diameters for dicot sunflowers and monocot maize plants in different conditions (free-standing and supported) and discuss the differences observed. Answer: The basal diameters of dicot sunflowers and monocot maize plants appear to be larger for free-standing plants compared to supported ones. For sunflower plants, the difference between the free-standing and supported basal diameters is 3.2 mm with a 95% confidence interval of (1.79, 4.61) mm. For maize plants, the difference between the free-standing and supported basal diameters is 1.6 mm with a 95% confidence interval of (0.8, 2.4) mm. This indicates that both types of plants have larger basal diameters when free-standing as opposed to supported, with sunflowers showing a greater difference in diameters compared to maize plants.

Step-by-step solution

Compute the difference in means for each plant

Using the given means in the table, we can calculate the difference in means by subtracting the mean of the supported plants from the mean of the free-standing plants for each type of plant: - For sunflowers: \(\Delta\mu_1 = 35.3 - 32.1 = 3.2\) - For maize: \(\Delta\mu_2 = 16.2 - 14.6 = 1.6\)

Calculate the 95% Confidence Interval for the differences

To calculate the 95% Confidence Interval (CI) for the differences for each type of plant, we'll use the formula for confidence intervals for two independent samples. As our sample size is 64, we can use a z-value of 1.96 for the 95% CI. Confidence Interval = \(\Delta\mu \pm z \times SE\) - For sunflowers: - CI = \(3.2 \pm 1.96 \times 0.72\) - CI = \((3.2 - 1.41, 3.2 + 1.41)\) - CI = \((1.79, 4.61)\) - For maize: - CI = \(1.6 \pm 1.96 \times 0.41\) - CI = \((1.6 - 0.8, 1.6 + 0.8)\) - CI = \((0.8, 2.4)\)

Write a paragraph with the conclusions

Based on the analysis, the basal diameters of the dicot sunflowers and monocot maize plants appear to be larger for the free-standing plants compared to the supported ones. For the sunflower plants, the difference between the free-standing and supported basal diameters was 3.2 mm, with a 95% confidence interval of (1.79, 4.61) mm. This means that we are 95% confident that the true difference in means for the sunflower plants lies between 1.79 mm and 4.61 mm. Similarly, for the maize plants, the difference between the free-standing and supported basal diameters was 1.6 mm, with a 95% confidence interval of (0.8, 2.4) mm. This indicates that we are 95% confident that the true difference in means for the maize plants lies between 0.8 mm and 2.4 mm.

Key concepts

Confidence Intervals

Confidence intervals are a fundamental concept in statistics. They offer a range of values that describe where we think the true value of a population parameter lies. When we say a 95% confidence interval, it means that if we repeated the experiment many times, 95% of the intervals calculated would contain the true mean.

To calculate a confidence interval, you need the sample mean and the standard error. The standard error measures the precision of your sample mean estimate and depends on both the standard deviation of the sample and the number of observations.

In the example with the sunflowers and maize, we used a fixed z-value of 1.96 (this value comes from the standard normal distribution and corresponds to the 95% confidence level). The calculation follows the formula: \[CI = \Delta \mu \pm z \times SE\] where \(\Delta \mu\) is the difference in sample means, \(z\) is the critical value (1.96 for 95% confidence), and \(SE\) is the standard error.

So, for our sunflowers, with a difference in means of 3.2 and a standard error of 0.72, the interval was calculated as \((1.79, 4.61)\). This means we are 95% sure that the true difference in basal diameter between free-standing and supported sunflowers falls somewhere between 1.79 mm and 4.61 mm.

Difference in Means

Understanding the difference in means is crucial when comparing two groups, like we did with free-standing and supported plants. The difference in means tells us how much one group differs from another on average.

In our exercise, we calculated these differences for both the sunflower and maize plants. For sunflowers, the difference was 3.2 mm, and for maize, it was 1.6 mm. These differences suggest that free-standing plants, on average, develop larger basal diameters compared to their supported counterparts.

This simple subtraction allows us to quantify how varying conditions impact the plants. It provides insight into whether changes like support or freedom in the wind affect plant growth.

Such information can be critical in agriculture and botany research. However, it's essential also to consider the variability and randomness in such samples, which we account for by using statistical methods like confidence intervals.

Experimental Design

Experimental design refers to the structured plan of an experiment, helping researchers maximize information while minimizing errors. An effective design includes elements such as randomization, control groups, and replication. In our exercise, the researchers compared two groups of sunflower and maize plants, distinguishing between those allowed to sway in the wind and those artificially supported.

By having two distinct groups, researchers can attribute observed differences specifically to the condition being tested (in this case, support structure) rather than other uncontrolled variables.

The use of 64 plants per group strengthens the reliability of the results. Larger sample sizes provide more data, reducing the impact of outliers and anomalies, leading to more generalizable outcomes.

Additionally, experimental design must account for potential sources of bias and errors, ensure consistency within conditions, and utilize statistical tools to derive valid conclusions. Through a well-structured design, such experiments can yield insights into the broader biological processes at play.