Evaluating Plant Fertilizers

1 Introduction

  Plant fertilizer provides nutrients to aid plant growth. Different formulas of nitrogen, potassium, and phosphorus have different effects on the growth of common grasses, such as fescue. In this study, three different fertilizer formulas were assessed for differences in fescue plant height.

2 Method

2.1 Objective.

  In this study, plant height was assessed for a common grass (fescue) based on the effects of three different fertilizer formulas which varied in ratios of nitrogen, potassium, and phosphate.

2.2 Data Collection.

   Each single seedling fescue plant was randomly assigned to a single group. The study involved four groups: three fertilizer formulas groups (F1, F2, and F3) and a group with no fertilizer (F0). At the end of the growth period, a single height measurement was taken for each plant. Fescue plant height was measured in centimeters. There were 24 observations collected. (The data was provided for Pennsylvania State University statistics projects.)

3 Statistical Analysis

  The study addressed whether there was a significant difference between the mean heights for the groups. It also addressed whether there were significant pairwise differences in plant height between each group. A completely randomized design was used, and differences in mean plant height were assessed using one-way ANOVA.

\[H_0: \mu_1= \mu_2 = \mu_3 = \mu_4\] \[ H_{alt}: \mbox{at least one mean is different}\] where the means \((\mu_{i..4})\) represent the means of the four different groups.

3.1 Descriptive Statistics.

  The summary table shows category counts for the four groups. The summary table also shows quantiles, minimum, maximum, arithmetic mean and medium for plant height (centimeters) for the entire data set. The data summary specified no empty data cells.
fertilizer height
F0:6 Min. :19.50
F1:6 1st Qu.:22.50
F2:6 Median :27.25
F3:6 Mean :26.17
NA 3rd Qu.:28.75
NA Max. :32.00

The summary table for the separate groups (F0, F1, F2, and F3) shows the arithmetic mean for plant height, medium plant height, variance, and standard deviation. The plot of group means is also shown.

fertilizer mean_height median_height var_height stdev_height
F0 21.00000 21.00 1.000000 1.000000
F1 28.60000 28.30 5.940000 2.437212
F2 25.86667 26.25 3.606667 1.899123
F3 29.20000 29.35 1.660000 1.288410

3.1.1 Boxplots.

  The boxplots for each group displayed no outliers.

3.1.2 Scatter Plot.

  A scatter plot displaying plant height (centimeters) on the y-axis and groups on the x-axis showed no significant outliers or high leverage observations.

3.2.1 ANOVA Assumptions.

  Each single seedling was randomly assigned to a single group, and the plant height measurements were obtained from independent observations. The residuals plots showed the mean residuals to be reasonably close to zero and evenly dispersed, supporting the assumption of homogeneity of variance. The QQ plot was approximately linear, supporting the assumption that error terms were close to the normal distribution.

4 Results

4.1 ANOVA

A one-way analysis of variance showed that there were significant differences in plant height means between the groups at the \(p<.05\) level, \(F(3,20)=27.46, p<.001\). Follow-up comparisons were performed.

##             Df Sum Sq Mean Sq F value   Pr(>F)    
## fertilizer   3 251.44   83.81   27.46 2.71e-07 ***
## Residuals   20  61.03    3.05                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##  (Intercept) fertilizerF1 fertilizerF2 fertilizerF3 
##    21.000000     7.600000     4.866667     8.200000

4.2 Post Hoc Test

Post hoc comparisons were performed using the Tukey HSD test. The results showed that all three fertilizer groups (F1, F2, and F3) were significantly higher than the group with no fertilizer (F0), at the \(p<.05\) level. The results also showed that formula F3 was significantly higher than formula F2 at the \(p<.05\) level. The mean plant height differences between formulas F1 and F2 were not significant at the p<.05 level. The mean height differences between formulas F1 and F3 were also not significant at the p<.05 level.

## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Multiple Comparisons of Means: Tukey Contrasts
## 
## 
## Fit: aov(formula = height ~ fertilizer)
## 
## Linear Hypotheses:
##              Estimate Std. Error t value Pr(>|t|)    
## F1 - F0 == 0    7.600      1.009   7.535   <0.001 ***
## F2 - F0 == 0    4.867      1.009   4.825   <0.001 ***
## F3 - F0 == 0    8.200      1.009   8.130   <0.001 ***
## F2 - F1 == 0   -2.733      1.009  -2.710    0.060 .  
## F3 - F1 == 0    0.600      1.009   0.595    0.932    
## F3 - F2 == 0    3.333      1.009   3.305    0.017 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)
##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = height ~ fertilizer)
## 
## $fertilizer
##            diff        lwr         upr     p adj
## F1-F0  7.600000  4.7770648 10.42293521 0.0000016
## F2-F0  4.866667  2.0437315  7.68960188 0.0005509
## F3-F0  8.200000  5.3770648 11.02293521 0.0000005
## F2-F1 -2.733333 -5.5562685  0.08960188 0.0598655
## F3-F1  0.600000 -2.2229352  3.42293521 0.9324380
## F3-F2  3.333333  0.5103981  6.15626854 0.0171033

5 Conclusions

  There were significant increases in plant growth when any of the three fertilizer formulas (F1, F2, and F3) were used in comparison to no fertilizer (F0). Additionally, formula F3 showed significantly higher growth than formula F2. With 95% confidence, we can assume: the mean height of fescue plants grown with fertilizer formula F1 will be between 4.78 to 10.42 centimeters higher than fescue grown without fertilizer; the mean height of fescue grown with formula F2 will be between 2.04 to 7.69 centimeters higher than fescue grown without fertilizer; the mean height of fescue grown with formula F3 will be between 5.38 to 11.02 centimeters higher than fescue grown without fertilizer; and the mean height of fescue plants grown with formula F3 will be between .51 to 6.16 centimeters higher than fescue plants grown with formula F2.

Data Source: From Penn State University stats project data.

03/2021