stats phase ii

7/31/2019 Stats Phase II

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Abstract

The formula c=d describes the relationship between a circular objects circumference

and its diameter, however, this paper seeks to prove that there is a constant

relationship between the circumference and diameter of round objects without the use

of said formula. It relies instead on measurements collected by college statistics

students and the analysis of the data collected to prove a direct correlation between the

circumference and diameter.

Stats 1510 Research Project! 2


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On the Relationship Between the Diameter and Circumference of Circular Objects

! The circumference, the enclosing boundary of a curved geometric figure, has

often been defined as the diameter of a curved figure multiplied by pi, a number

approximating to about 3.14. Many students have accepted this formula as law without

bothering to check the facts themselves, however, in the following experiment, I will

seek to prove that the circumference can accurately be calculated by multiplying the

diameter by pi.

Method

Participants

! The following technique was used by an online college statistics class to

measure the circumference and diameter of 392 objects. Once the data was collected

by the students, it was submitted to the moodle website where the professor compiled

the information onto a spreadsheet.

Measures

! In this experiment, objects were measured using a piece of string and a number

of rulers. The string was placed along the center of the face of the circular object like so:



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(Please note that tape was not used in the measurement process, but for photographing

purposes only.) The string was then marked with a pencil where it met the edges of the

face of the circular object. The space between these marks was measured with a ruler

to record the diameter of the object as pictured below.



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The same general process was used to measure the circumference of the circular

object. The only difference being that the string was wrapped around the body of the

object like so:



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Again, the tape was only used for photographing purposes. After the string was marked

on the places it overlapped, the space between the two marks was measured as before.

Participants

This technique was used by an online college statistics class to measure the

circumference and diameter of 392 objects. Once the data was collected by the

students, it was submitted to the moodle website where the professor compiled the

information onto a spreadsheet.

Procedure:

The data associated with 25 objects was randomly selected using a TI84 calculator s

randInt application (randInt- >0,392, 25), and recorded



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in the following table. Also recorded was the Pi (circumference/diameter) of each object.

Results

Table 1

Experimental Data

Item # Object Circumference Diameter Pi

1 Cup 262 82.5 3.1758

2 Mason Jar Lid 8.75 2.5 3.5

3 Green Ball 6.2 1.6 3.875

4 Quarter 3.375 0.938 3.5981

5 Clock Face 73 23.1 3.1602

6 Baseball 30 6 5

7 Campbells Soup 8.5 2.625 3.2381

8 Vase 19.25 6 3.2083

9 Blue Bowl 11.5 5.75 2

10 CD 18 4.75 3.7895

11 Toilet Paper Roll 27.946 2.55 10.959

12 Yo-Yo 8 2.8 2.8571

13 Drum 37.75 12.125 3.1134

14 Tuna Can 10.625 3.375 3.1481

15 Trash Can 27.5 8.75 3.1429

16 Poker Chip 122.5 39 3.141

17 AA Battery 43.8 14 3.1286

18 Corn Tortilla 45.7 14 3.2643

19 Ping Pong Ball 5 2 2.5

20 Canned Peas 9.25 3 3.0833



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21 PX Bottle 4.25 1.25 3.4

22 Wall Clock 31.5 9 3.5

23 Ranch Beans 283 88.9 3.1834

24 Mirror 44.4 36.25 1.2248

25 Stove Knob 2.5 1 2.5

This data can be assumed to be representative of the population (the other objects

measured by the students), because it was obtained through the randomizing features

of the TI-84 calculator, which possesses no capacity for bias or human error. Using 1-

VarStats on the columns consisting of the measurements for Circumference, Diameter,

and Pi, the following data was calculated.

Table 2

Statistics Related to the Circumference, Diameter, and Pi Calculation.

Mean Median SD Q1 Q3 Min. Max.

Circumference

Diameter

Pi

47.18 19.25 73 8.25 44.1 2.5 283

14.95 5.75 24 2.525 14 0.938 88.9

3.468 3.1758 1.7 3.0984 3.5 1.2248 10.9592

Although the statistics pertaining to both the circumference and the diameter of the data

fluctuates drastically, those related to Pi (circumference/diameter) stay relatively

constant. The standard deviation from the mean is also quite small, suggesting that the

relationship between the circumference and the diameter of each object holds fairly

constant as well.



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! In order to test the hypothesis that there is a constant relationship between the

circumference and diameter of round objects, a linear regression was performed on the

circumference and diameter data located in Table 1. The steps and procedures used are

as follows:

1. A scatter plot was constructed with the circumference listed on the x-axis and the

diameter listed on the y-axis. (See Scatter Plot 1 in the Index for graph.) With the

exception of one outlier, the data appeared to exhibit a strong, positive linear trend.

2. The two assumptions needed to calculate Pearsons correlation were then

addressed:

a. Both variables were randomly selected from the populations they represented

through the use of the RandInt function on the calculator.

b. Together the variables came from a bivariate normal population. This assumption

was met by the linear trend exhibited by the scatter plot.

3. Using a calculator, the sample correlation was calculated and found to be 0.9793,

23.203 reenforcing a strong, positive linear trend.

4. A complete hypothesis test was then constructed and yielded a p value of

approximately zero, leading to the conclusion that the null hypothesis could be

rejected, that is Ho: =0

Discussion

! Upon examination of these statistics, it is reasonable to conclude that there is a

definite relationship between the circumference of a curved geometrical figure and its

diameter. According to common knowledge, this relationship can be found in the form of

pi, a number approximating to about 3.14, however, the data shown in Table 2. has a



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slightly higher median value of 3.18, which could easily be the result of human error in

the measurement process. For example, the students might have improperly recorded

the results of either their circumference or diameter measurement, or even measured

incorrectly. It is also important to address the outlier mentioned in the previous section.

This value came from a student who measured a toilet paper roll with a circumference

of 27.946 and a diameter of 2.55, resulting in a pi value of 10.959. These values could

have been the result of human error as well, but were still included in all of the

calculations listed in the Results section of this paper, resulting in slightly skewed data.

Index

Table 4.

Sample Data


1 Cup 262 82.5 3.1758

2 Mason Jar Lid 8.75 2.5 3.5

3 Green Ball 6.2 1.6 3.875

4 Quarter 3.375 0.938 3.5981

5 Clock Face 73 23.1 3.1602

6 Baseball 30 6 5

7 Campbells Soup 8.5 2.625 3.2381

8 Vase 19.25 6 3.2083

9 Blue Bowl 11.5 5.75 2

10 CD 18 4.75 3.7895

11 Toilet Paper Roll 27.946 2.55 10.959

12 Yo-Yo 8 2.8 2.8571



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13 Drum 37.75 12.125 3.1134

14 Tuna Can 10.625 3.375 3.1481

15 Trash Can 27.5 8.75 3.1429

16 Poker Chip 122.5 39 3.141

17 AA Battery 43.8 14 3.1286

18 Corn Tortilla 45.7 14 3.2643

19 Ping Pong Ball 5 2 2.5

20 Canned Peas 9.25 3 3.0833

21 PX Bottle 4.25 1.25 3.4

22 Wall Clock 31.5 9 3.5

23 Ranch Beans 283 88.9 3.1834

24 Mirror 44.4 36.25 1.2248

25 Stove Knob 2.5 1 2.5



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Table 3.

Hypothesis Test

1. 5. p=0

2. Ho: =0 Ha:0

6. reject the nullhypothesis.

3. t-distribution 7. There is enough

4. The data is normalSee Scatter Plot 1.

there is a correlationbetween the circumferenceand the diameter of acircular object.

Scatter Plot 1.

Circumference vs. Diameter.


0

22.5

45

67.5

90

0 75 150 225 300

Circumference vs. Diameter


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My email crashed. Please accept this link as a substitute for my rough draft as is

contains most of the information needed.

http://podcasts.taftcollege.edu/podcasts/stat1510/students/exams/summer12/

Napier4109.m4v


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