Acorns and range might be directly correlated and the validity of acorn sizes actually affecting region ranges is questionable so we used linear regression analysis to determine the rationality. Excluding the leverage points, the p value and coefficient of determination values were in favor of the alternative hypothesis, larger acorn sizes expands region ranges, so we reject the null hypothesis. The main complication within the data set were the leverage points, but once they were disqualified from the data we were able to get a more accurate study.
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Acorns, also known as oaknuts, are the nuts of oak trees that have a single seed inside of them and over 100 animals in the US consume these nuts. Since the seed lies within the acorn in order for another oak tree to be planted they rely on the animals to disperse them around the environment. Usually only small animals eat the acorns, so how are other oak trees sprouting up miles away from the parent tree? The relationship in this linear regression analysis is between acorn size and the range of the region they are produced at. It’s suggested that region’s with greater ranges tend to produce bigger acorns that will attract larger animals who are able to carry the acorns and drop them further away from its parent tree thus creating a larger range. To determine the probability that this might be the case we must first state the null and alternative hypothesis. The null hypothesis is that the acorn size does not have any direct correlation with the range of the region, the beta will be equal to zero. The alternative hypothesis is the bigger the acorn size the larger the region area becomes, the beta will be greater than zero. Our data set comes from 11 out of 50 species of oak tree and all regions are from California. X, the explanatory variable is the acorn size, and Y, the response variable is the range of how far away a new oak tree can grow. They have a positive correlation. With this information we will be able to find beta, the level of how much the range changes based on the size of the acorn.(at what percent are the sizes of the acorns effecting the region range.)
Through further analysis using scatter plots and the equation we got for y=a+bx, y=517.873+13.873x, we were able to find the beta, p value, and coefficient of determination of the data set. For the P-value we obtained the value .5519. P value is a summary of the data that gives us an insight on the possible outcome of the research and based on the unusual measurement of p-value it can trigger the rejection of the null hypothesis. Let’s say the significance level of this analysis is .05. In that case, an extremely small p-value is very strong evidence against the H0, but for the original data set we do not reject the null hypothesis since the p value is greater than the level of significance. For the coefficient of determination (r^2) value you want the opposite, the higher the value of r^2 the stronger evidence you have to determine if the x variable has any effect on the y variable. The value for r^2 is .0407. The number is very low and does not give us strong evidence x affects y.
If we look back at the data set, there are two leverage points, sizes 17.1 and 7.1. We’ll be removing 17.1 because the x variable, acorn size, is way too high compared to the other number in the data set. This particular acorn is grown on a Quercus Chrysolepis which are grown in areas with high amounts of rainfall that would have a high positive influence in acorn development. 7.1 is being removed as well since the its y variable, region range, its parent tree sits on is only 13 km2x100 since it’s on an island, this limits its growth and does not fit in with the data set. Having these leverage points could possibly weaken the data values dramatically that could be more accurate without them. Once we remove the leverage points from the data set the new equation we’ll be using is y=318.695+98.445x. The p value and r^2 value completely changes and now favors the alternative hypothesis. The p-value is now 0.0205 and r^2 is .5591. Still staying with a 0.05 significance level, p value is now smaller than 0.05 which favors the alternative hypothesis that the bigger the acorn size the larger the region area is.
To continue, we’re going to take a closer look at the Coefficient of Determination (r^2). The data we’re going to focus on now is the graph and set without the leverage points which received the value .5591 for r^2. The higher the value the stronger the evidence will be in favor of x affecting y. .5591 means roughly 56% of the range can be explained by the size of the acorn, but 45% is a huge chunk that can’t be explained. There are many confounding variables that can explain that 45%, for example a catastrophic event within the animal ecosystem. Jays and squirrels are the animals that mostly eat acorns, squirrels can carry multiple acorns at once and jays can only carry one or eat it on site. In an animal community each animal plays a key role in population control with animals and plants, so if there was an accidental spike of an animal the balance will be off. Let’s say we have a high region range of 800 km2x100 with an average amount of oak trees and the coyote population spiked then hunted almost all of the squirrels. This means the animal that had the highest consumption rates of acorns is now little to none and the only animal really eating the acorns are jays. Since jays are small and can’t acquire many acorns at once, the amount and sizes of the acorns will rise since it will not be picked off the tree as often. If the analysis is done within this area, the misleading conclusion would be that the acorn sizes affect the range.
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As we go on, even if the confounding variable isn’t as big as the population event, a small variable can compromise the analysis. Like we saw with acorn 17.1 and it’s y-variable, we talk about the region and range, but the weather and high amounts of rain activity could positively help acorns grow in size regardless of range. Despite the confounding variables, the data set without the outliers showed significant data favoring the null hypothesis that acorn size does affect range. The graphs residual analysis concurred this accusation since the graphs are normally distributed and shows a strong linear relationship.
In conclusion, we decide that the bigger the acorn size the bigger the region becomes and reject the null hypothesis that acorn size has no correlation to range. This is decided without the leverage points since they can dramatically skew the data and give you inaccurate results. Always make sure your data isn’t skewed or inaccurate so the end results are clear. Referring back to Wheelan, you don’t want garbage in or your output will be garbage as well. The results favored that the x value affected y value, but there are many confounding variables that can explain y value. Hence why we don’t state the theory to be true, we just either chose to reject or not to reject.
- Quercus Chrysolepis, www.fs.fed.us/database/feis/plants/tree/quechr/all.html#46.
- Wheelan, Charles J. Naked Statistics: Stripping the Dread from the Data. W.W. Norton & Company, 2014.
Aizen, Marcelo A., and William A. Patterson. “Acorn Size and Geographical Range in the North American Oaks (Quercus L.).” Journal of Biogeography, vol. 17, no. 3, 1990, p. 327., doi:10.2307/2845128.
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