Question

Radish seedlings were randomly distributed and then grown under four conditions: natural lighting, darkness, regular (incandescent) light bulb, and full spectrum light bulb (providing ultraviolet lighting). Heights of the seedlings were measured after 9 days. The data are summarized below. \(^{10}\) $$ \begin{array}{|lcccc|} \hline & \text { Natural } & \text { Dark } & \text { Regular bulb } & \text { Full spectrum } \\ \hline \text { Mean } & 5.72 & 9.06 & 1.40 & 5.16 \\ \text { SD } & 2.19 & 4.09 & 1.87 & 2.44 \\ n & 20 & 25 & 20 & 24 \\ \hline \end{array} $$ (a) State in words the null hypothesis that is tested using ANOVA here. (b) The \(P\) -value for the \(F\) test is \(2.9 \times 10^{-12}\). What is your conclusion regarding the null hypothesis? (c) Is it proper to draw a cause-effect conclusion from these data? Why or why not?

Short answer

The null hypothesis states that there's no significant difference in seedling heights across the four growth conditions. Based on the P-value of 2.9 x 10^-12, the null hypothesis is rejected, indicating significant differences in mean heights. However, cause-effect conclusions cannot be drawn from the data without further controlled experimental investigations.

Step-by-step solution

Understanding the Null Hypothesis

The null hypothesis for an Analysis of Variance (ANOVA) test states that all groups have the same population mean. In this context, the null hypothesis suggests that there is no significant difference in the mean heights of radish seedlings grown under different conditions, which means the mean heights for seedlings grown in natural lighting, darkness, regular light bulb, and full spectrum light bulb should be equal.

Analyzing the P-Value to Draw Conclusions

When evaluating ANOVA results, if the P-value is less than the significance level (typically 0.05), we reject the null hypothesis. Here, the P-value is very small (2.9 x 10^-12), which means it's significantly less than 0.05 and we can reject the null hypothesis with high confidence.

Addressing Cause-Effect Conclusions

While ANOVA can show there are differences in seedling growth across various light conditions, it does not establish a cause-effect relationship. The study is observational and the differences could be influenced by other variables not controlled for. Therefore, without a controlled experimental design, it would not be proper to conclusively draw cause-effect conclusions from these data.

Key concepts

Null Hypothesis in ANOVA

The null hypothesis in an Analysis of Variance (ANOVA) is a statistical assertion that aims to establish a baseline for evaluating whether observed differences in data are of any real significance. In biological research, such as the case with radish seedlings' growth under varying light conditions, the null hypothesis posits that no true difference in mean heights exists across the groups (natural light, darkness, regular bulb, full spectrum bulb).

When researchers assert this hypothesis, they essentially claim that any observed variation in seedling height is due to random chance rather than the effect of the different lighting conditions. If the null hypothesis is disproven, it implies that at least one of the lighting conditions has a significant effect on the growth of the seedlings. To test this, the ANOVA checks for differences among the means of the various groups and determines whether these discrepancies can be attributed to experimental factors or just random variability.

P-value Interpretation

The P-value is a crucial aspect of hypothesis testing in statistics, encapsulating the probability of obtaining a result at least as extreme as the one observed during the experiment, under the presumption that the null hypothesis is true. For the ANOVA conducted in the radish seedling experiment, a P-value of 2.9 x 10^-12 indicates an exceedingly small chance that the observed differences in seedling growth are due to random chance alone.

This incredibly small P-value leads us to reject the null hypothesis, supporting the assertion that different lighting conditions do indeed have varying impacts on seedling height. It's common convention in scientific research to use a threshold such as 0.05 as the significance level. Since the P-value here is far below 0.05, the likelihood of the null hypothesis being correct (all groups having equal mean heights) is exceptionally slim, leading researchers to conclude that the type of lighting is significantly associated with the growth of the seedlings.

Cause-Effect Relationship in Experiments

Determining a cause-effect relationship is a goal in many experiments; however, demonstrating this relationship goes beyond merely finding statistical differences. In the case of our radish seedlings study, while the ANOVA can indicate that various lighting conditions lead to different growth outcomes, it does not inherently explain why these differences occur. To establish cause and effect, one would need a controlled experimental design that systematically manipulates the light conditions while holding all other potential influencing factors constant.

Furthermore, an experiment must have replication, randomization, and control to attribute changes in the dependent variable (seedling height) directly to the independent variable (light conditions). Since the study described was observational, there could be other unaccounted factors contributing to the differences in seedling growth, and thus, cautious interpretation of the results is warranted. Researchers must, therefore, avoid making unwarranted cause-effect claims without further experimental evidence that eliminates alternative explanations for the observed results.