Predicting Rate of Falling Apples from Beginning of Week Three to the Beginning of Week Four

by Science Student

California State Polytechnic University

Pomona, CA

2003

(Where is the abstract?)

(Where is the table of contents?)

Introduction

(Poorly written intro statement…don’t start with “the goal is”)

The goal is to determine the rate that apples drop from the apple trees during harvest season at the Johnny Apple Core Farm. Previous background research conducted at the Johnny Apple Core Farm suggests that until 2/3 of the apples have fallen from the tree, the number of apples dropping from the tree increases as a square function, i.e. on day 4 (from the start of harvest season) about 16 apples will drop from a tree and on day 6 about 36 apples will fall (Apple Science, 2002) as shown in Picture 1. At the time that 1/3 of the apples are remaining the drop rate abruptly tapers off in a negative linear fashion until all the apples have dropped from the tree. (Why do we want to know the rate of falling apples? This need for the rate should be included in the introduction with the “goal” or purpose.)

Picture 1

-New Section-

Preliminary research conducted on the Johnny Apple Core Farm has determined an increase in apple collection efficiency when the apples are allowed to naturally drop from the trees (Saving Apples, 2003.) The average tree of the ten of interest supports 1000 apples at the beginning of harvest season; where harvest season is defined as the first day that at least one apple drops per day until the apple trees are bare. Once the apples lay on the ground the use of a ladder or a lift to remove the apples from the tree is unnecessary. This method of collecting apples is against popular opinion (Popular Apple News, 1999.)

The collected number of apples is documented for the first two weeks of harvest season. Each day the apples are collected and counted at 8 AM, 1 PM and 7 PM. The analysis of this data should be about to predict the next week of apple collection.

(Unbeknownst to Johnny the local squirrels are also waiting for the apples to begin to drop. There are 10 squirrels. Each squirrel gathers their bounty of one apple early in the morning..)

(What is this??? Why is it italicized? Is this determined by analyzing the data and determining there is a reducing factor? Determining this “reducing factor” is an experiment in itself; thus should be included in the Discussion section, for further research.

-New Section-

Table 1 reports the accumulated 8 am, 1 pm, and 7 pm totals for each day. Graph 1 demonstrates that the data is not linear. The third column of Table 1 is the square root of the number of apples collected on each day shown in Graph II.

Table 1
Apples Collected Each Day
Day / # of apple collected / Square root of apples
1
2
3
4
5
6
7
8
9
10
11
12
13
14 / 6
14
27
38
50
75
90
110
135
160
190
215
250
270 / 2.4
3.7
5.2
6.2
7.1
8.7
9.5
10.5
11.6
12.6
13.8
14.7
15.8
16.4

(Why does the table have raw data and calculated values? Raw data if included should be included in the Appendix section. Analyzed data should be included at this point.)

Graph II, the square root of the apples collected versus the day collection occurred, corroborates the literature review that the number of apples collected each day is directly proportional to the square of the day on which the collection occurred. (So why is Graph 1 included? Graph 1 should only be included is if the abstract included something like “commonly incorrect determinations are numerous. These incorrect predictions are shown below to be invalid along with…)

The slope of Graph 2, 1.07 sqrt Apples / Day, is the rate at which apples fall during the first two weeks. The y-intercept is 1.81 sqrt Apples or 3.3 apples. Forcing the y value in the y-intercept formula

Eq 1: y = mx + b,

and solving for x allows the determination of the x-intercept which yields a value of -1.69 days.

(Never force anything in a graph. If the x-int is desired the correct method is y = m x + b; the value for y is zero as the line passes through the x-axis, which give the following equation

0 = m x-int + b or x-int = -b/m.)

New Section

The x-intercept of -1.69 days leads to a conclusion that somehow apples are disappearing before morning collections. Although the data is highly correlated with a correlation coefficient of 0.9988 which accounts for 99.77% of the variability. (Not a complete sentence.)

(Is this a new paragraph?)

If square root of apples collected represents the data at the lower bounds, this will be significantly influenced by a constant removal of apples. As a result a new graph is required that omits the data at the lower bounds (this is data that is most influenced by a probable systematic error).

(Must explain why, “most likely”. What does this mean?)

Graph 3 (data removed with the greatest fractional uncertainties and most likely to be influenced by systematic error) yields a new y-intercept of 2.11 and a slope of 1.05, which corresponds to an x-intercept of exactly -2.00 days. The conclusion is that the documented first day of apple collection is not actually the first day, but the third day after apples began dropping off the tree! The entire x-axis is shifted by 2 days to the left by systematic error show in the below formula.

Eq 2: Original Day + x-intercept = True Day after Apples Began Dropping.

The y-intercept of 2.1, the square root of the number of apples on what was believed to have been Day 0, is now known to be Day 2.

Correspondingly, Day 1 should have had approximately 1 apple dropping from the tree, Day 3 about 9 apples dropping from the tree, and Day 4 about 16 apples dropping from the tree.

With Graph 3, it’s been demonstrated that an unknown factor was removing up to approximately 10 apples every day before 8 AM. The conclusion is that the early squirrel gets the apple…about 10 apples every day. (Placing humor may sound like a good thing, but humor doesn’t really belong in a journal. Why? Do you want to read the least amount for find answers? So does everyone else.)

Appendix A: Raw Data

Days Johnny / True Days / Apples2 / Apples2 - 10 / (Apples2 - 10)1/2 / Apples2 - 10
collected / w/ uncertainty / w/ uncertainty
apples / 1 / 1
2 / 4
3 / 9 / 0
1 / 4 / 16 / 6 / 6
2 / 5 / 25 / 15 / 14
3 / 6 / 36 / 26 / 5.2 / 27
4 / 7 / 49 / 39 / 6.2 / 38
5 / 8 / 64 / 54 / 7.1 / 50
6 / 9 / 81 / 71 / 8.7 / 75
7 / 10 / 100 / 90 / 9.5 / 90
8 / 11 / 121 / 111 / 10.5 / 110
9 / 12 / 144 / 134 / 11.6 / 135
10 / 13 / 169 / 159 / 12.6 / 160
11 / 14 / 196 / 186 / 13.8 / 190
12 / 15 / 225 / 215 / 14.7 / 215
13 / 16 / 256 / 246 / 15.8 / 250
14 / 17 / 289 / 279 / 16.4 / 270
15 / 18 / 324 / 314 / 17.7 / 315

As you can see…no calculation were actually included in the journal article.

This is a very rough journal article.

Please analyze both x and y axes. Graphs rarely go through the origin due to systematic error, friction, etc.

Remember y = mx + b; so if you know the y-intercept and the slope…just plug in a zero value for y and you’ll get the x-intercept.

Never use data points to calculate a slope.

If using MS Excel, always make scatter plots…after the chart created…right click on any data point and Add Trendline

Never say HUMAN ERROR. Always be specific in your explanations. That doesn’t explain anything explain exactly what is meant by human error.

Never make a general statement…it’s worse on your grade to make a general statement than no statement at all.

NEVER connect DOTS

NEVER use individual data points for ANYTHING.