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Elementary Statistics

Hypothesis Testing the Golden Ratio

The Golden Ratio, sometimes referred to as Phi (, is a ratio between two lengths A and B that yield the exact number which is irrational and is approximately We will use 1.62 as our approximation for the purposes of this hypothesis test. Many claims have been made that the golden ratio exists in nature both in living and non-living things.

For example, the length of a human arm from elbow to wrist compared to the length of the hand from wrist to tip of middle finger. In this example, the length from elbow to wrist would be length A and the length from wrist to middle finger would be length B.

To test a claim such as this, suppose we collected a large data set and fond the mean average ratio for all the ratios collected to be 1.71. This is indeed close to 1.62, but do we then agree with the claim that human arms and hands grow in an approximate golden ratio? What if the original claim was that this human ratio was which is approximately ? Do our findings support this claim? This example shows why we never, ever support the null hypothesis with our findings. We may only reject it due to strong evidence to the contrary, or fail to reject it due to insufficient evidence to the contrary.

For this example we would have the following:

Notice that we are using the claim that the ratio of arm length to hand length is in the golden ratio as our null hypothesis. This may seem strange, but it is the only way to proceed since we would have no other default position for the null hypothesis. Therefore our test may only reject this claim or fail to find sufficient evidence against it (which is not the same as supporting the claim).

For this project we will not use human ratios, but rather natural objects that are readily available to you. Search out items that you have access to that are abundant enough to enable you to find a random sample of 30 or more that represents a much larger population. Measure two different lengths on the object and compute the ratio (long/short). Try different objects until you get initial measurements that yield a ratio very close to 1.62.

Example: Suppose I walk around my backyard and measure the width and length of an apple leaf and find the ratio is only 1.2. This is not even close to 1.6 so I move on to an oak leaf. The ratio is 1.4, again not close enough so I move on to an ivy leaf. The initial ratio (from one leaf) is 1.64, eureka! This is close enough to XXXXXXXXXXFrom this initial finding I settle on using ivy leaves for this test. Thus, my null hypothesis is that the mean ratio of the length to width of all ivy leaves is 1.62 (our approximation of the golden ratio). Next, I come up with a way to randomly select 30 or more ivy leaves of the same species (different plants if possible). I create a data table of all the measurements and ratios I calculated for each leaf. I do not measure each leaf as I go and only include those that are close to 1.62. I collect 30 or more and then measure them all and whatever ratio I find is what gets entered into the data table.

Leaf #

Side A (long side)

Side B (short side)

Ratio A/B

1

3.10

1.90

1.63

2

3.41

1.93

1.77

29

2.54

1.41

1.80

30

3.62

2.23

1.62

I now use the Ratio A/B column to calculate all of my summary statistics, such as mean, median, standard deviation, Q1, Q3 and so on. I use online tools such as StatKey to evaluate the claim (more on that later) and decide if I shall reject the claim or fail to reject the claim due to insufficient evidence.

Your Project: Using the example above, locate an item to generate a claim and then test the claim using statistical tools. What follows is a scaffolding of how to arrange your findings for your formal report that is to be turned in as a word.doc or PDF. Do not write in bullet point format.

The Formal Report: Each student must turn in his or her own unique report. I would like to read your thoughts and in your words. You are allowed and encouraged to discuss your ideas with classmates, the math tutoring center, the writing center, and myself during office hours. Please DO NOT email me any rough drafts. I also recommend that you never email a classmate your report file or a link to a Google drive. The report will be 12pt type, Arial or Calibri font, and 1.15 to 1.25 line spacing. Use the bolded headings below for your report, in the exact order displayed. Do not use bullet points in your writing. Embed all tables and graphs in-line (meaning place them near the text where they are referenced). Use screen capture (image clipping) for graphical displays from StatKey. Size the images to make them readable. There is no minimum or maximum length to the report. It should be as long as is needed. I highly recommend that you visit the writing center and allow someone to review your work for clarity.

Abstract: This is a brief summary of your findings. Include all metrics such as p-values, t-statistics and confidence intervals and what they mean in the context of the investigation (briefly). You will typically write this last even though it comes first in the report.

Example:

In regards to the ratio of the length vs. the width of the common ivy leaf (Hedera), we found statistically significant evidence, p-value=0.001, that this ratio is different from 1.62. Additionally, we estimate the true parameter of this ratio to be between …

More can be said here, but should be 3 to 6 sentences.

Introduction: Here is where you introduce the reader to the study. Give some background on the golden ratio. State the statistical question of interest. What are you investigating? What parameter are you trying to estimate? What methods did you use (box plot, p-value…)? What randomization technique did you use when collecting data (this is important)? Include one image of the overall objects you are colleting. Example: If you are collecting ivy, take a picture of the overall ivy bush and include it in the report. Also include an image, or two, of how you are measuring the length and width of the objects. Example: If you are measuring ivy leaves, you might show a picture of an ivy leaf flat on a table with a ruler indicating what you are considering to be length and width.

Histogram and Boxplot Analysis Using Summary Statistics:

· Using StatKey (or other online stats packages), create a histogram and boxplot of your ratio column. Using One Quantitative Variable tab, you can enter your own data into StatKey under “edit data” or “upload file”. Use screen capture (not a photograph of the screen) to clip the histogram and box plot from StatKey and place them here.

Example: Here is the histogram for an underlying data set.

(This example is not about the golden ratio):

· Describe the shape and symmetry of the histogram and which way (left/right) the mean is skewed away from the median. Next, measure the relative skewing by using the standard deviation (from your data set) as a ruler. Meaning, how many standard deviations fit inside the absolute distance between mean and the median?

Example: Here is the boxplot for an underlying data set.

(This example is not about the golden ratio):

· Describing the variance of the data set: We want to evaluate each quartile for variance. State the width of each quartile in absolute value. Example: Quartile 1 is the lower 25% of the data set. Thus the width of quartile 1 is . The width of quartile 2 would be and so on. Note that is referred to as the Median. You should have four measures, one for the width of each quartile. You do not need to describe the calculations, but instead just least these measures in order and which quartile they represent.

· Describe the relative variance of the middle 50% (IQR).

What do you think this number is describing? What number do you think is a large, or small, amount of variance?

Hypothesis Test, Confidence Interval and Conclusions:

· State the hypotheses.

· Calculate the theoretic Standard Error (S.E.) using your data set (6.2 in text).

· Calculate the t-statistic (think z score).

· Using the theoretic t-distribution in StatKey, find the p-value for your t-statistic (and therefore of the mean for your ratio column).

· Describe in detail in the context of the investigation what the p-value means. (Example using human body temp: If we assume the average human body temperature is , we find the probability of our statistic occurring by random natural variance is …)

· State a conclusion about the hypotheses using a significance level of

· Create a 95% confidence interval (using the SE above) for the true parameter and describe specifically what you are 95% confident about.

Bias and Critiques: In this section you should critique your own work. How well do you think your randomization scheme worked in preventing bias? Describe at least one form of bias in your data set (there will be at least one). What would you do to improve it next time? How accurately do you think your measurements were? What could you do to improve this? Do you believe your conclusions are reliable based on the underlying data set? And anything else you would like to improve or do differently in the future.

Personal Reflections: (there is no right or wrong here, except to leave it blank)

○ What parts of this project were difficult for you? Were there technology issues that got in your way? Did you find the writing center/math tutoring helpful (if used)?

○ What did this project help you understand about these statistical tools?

○ What do you think this project measures about you and your understanding of the material and in what way is this measurement different from a timed exam (is this more work)?

· Do you prefer timed exams?

○ How could this project be improved?

Data Set: Embed a table containing all of your measurements and data.

***Notes to you:

· DO NOT write in bullet point form. Write in the fashion the professional papers we have viewed.

· There is no minimum length. The report should be as long as is needed to discuss all of the above.

· All of my example images above are from StatKey and are NOT about the golden ratio. They are from varied data sets and are there as examples of how to place screen captures in your document.

· DO NOT send your final project to me via email or a Googl.doc link. The final report needs to be a word.doc or a PDF uploaded to Canvas in the link provided for the project. (You can work on this in Google.doc if you wish, but you will need to download as a word.doc when finished and then upload to Canvas link.)

· Feel free to discuss with tutors, fellow classmates and myself. Every project must use a unique data set that is collected by you, thus everyone’s findings will naturally be slightly different.

· Success on this project is not about finding the golden ratio. Success is about analyzing your results and meaningfully describing them with relevant statistics.

Hypothesis Testing the Golden Ratio

The Golden Ratio, sometimes referred to as Phi (, is a ratio between two lengths A and B that yield the exact number which is irrational and is approximately We will use 1.62 as our approximation for the purposes of this hypothesis test. Many claims have been made that the golden ratio exists in nature both in living and non-living things.

For example, the length of a human arm from elbow to wrist compared to the length of the hand from wrist to tip of middle finger. In this example, the length from elbow to wrist would be length A and the length from wrist to middle finger would be length B.

To test a claim such as this, suppose we collected a large data set and fond the mean average ratio for all the ratios collected to be 1.71. This is indeed close to 1.62, but do we then agree with the claim that human arms and hands grow in an approximate golden ratio? What if the original claim was that this human ratio was which is approximately ? Do our findings support this claim? This example shows why we never, ever support the null hypothesis with our findings. We may only reject it due to strong evidence to the contrary, or fail to reject it due to insufficient evidence to the contrary.

For this example we would have the following:

Notice that we are using the claim that the ratio of arm length to hand length is in the golden ratio as our null hypothesis. This may seem strange, but it is the only way to proceed since we would have no other default position for the null hypothesis. Therefore our test may only reject this claim or fail to find sufficient evidence against it (which is not the same as supporting the claim).

For this project we will not use human ratios, but rather natural objects that are readily available to you. Search out items that you have access to that are abundant enough to enable you to find a random sample of 30 or more that represents a much larger population. Measure two different lengths on the object and compute the ratio (long/short). Try different objects until you get initial measurements that yield a ratio very close to 1.62.

Example: Suppose I walk around my backyard and measure the width and length of an apple leaf and find the ratio is only 1.2. This is not even close to 1.6 so I move on to an oak leaf. The ratio is 1.4, again not close enough so I move on to an ivy leaf. The initial ratio (from one leaf) is 1.64, eureka! This is close enough to XXXXXXXXXXFrom this initial finding I settle on using ivy leaves for this test. Thus, my null hypothesis is that the mean ratio of the length to width of all ivy leaves is 1.62 (our approximation of the golden ratio). Next, I come up with a way to randomly select 30 or more ivy leaves of the same species (different plants if possible). I create a data table of all the measurements and ratios I calculated for each leaf. I do not measure each leaf as I go and only include those that are close to 1.62. I collect 30 or more and then measure them all and whatever ratio I find is what gets entered into the data table.

Leaf #

Side A (long side)

Side B (short side)

Ratio A/B

1

3.10

1.90

1.63

2

3.41

1.93

1.77

29

2.54

1.41

1.80

30

3.62

2.23

1.62

I now use the Ratio A/B column to calculate all of my summary statistics, such as mean, median, standard deviation, Q1, Q3 and so on. I use online tools such as StatKey to evaluate the claim (more on that later) and decide if I shall reject the claim or fail to reject the claim due to insufficient evidence.

Your Project: Using the example above, locate an item to generate a claim and then test the claim using statistical tools. What follows is a scaffolding of how to arrange your findings for your formal report that is to be turned in as a word.doc or PDF. Do not write in bullet point format.

The Formal Report: Each student must turn in his or her own unique report. I would like to read your thoughts and in your words. You are allowed and encouraged to discuss your ideas with classmates, the math tutoring center, the writing center, and myself during office hours. Please DO NOT email me any rough drafts. I also recommend that you never email a classmate your report file or a link to a Google drive. The report will be 12pt type, Arial or Calibri font, and 1.15 to 1.25 line spacing. Use the bolded headings below for your report, in the exact order displayed. Do not use bullet points in your writing. Embed all tables and graphs in-line (meaning place them near the text where they are referenced). Use screen capture (image clipping) for graphical displays from StatKey. Size the images to make them readable. There is no minimum or maximum length to the report. It should be as long as is needed. I highly recommend that you visit the writing center and allow someone to review your work for clarity.

Abstract: This is a brief summary of your findings. Include all metrics such as p-values, t-statistics and confidence intervals and what they mean in the context of the investigation (briefly). You will typically write this last even though it comes first in the report.

Example:

In regards to the ratio of the length vs. the width of the common ivy leaf (Hedera), we found statistically significant evidence, p-value=0.001, that this ratio is different from 1.62. Additionally, we estimate the true parameter of this ratio to be between …

More can be said here, but should be 3 to 6 sentences.

Introduction: Here is where you introduce the reader to the study. Give some background on the golden ratio. State the statistical question of interest. What are you investigating? What parameter are you trying to estimate? What methods did you use (box plot, p-value…)? What randomization technique did you use when collecting data (this is important)? Include one image of the overall objects you are colleting. Example: If you are collecting ivy, take a picture of the overall ivy bush and include it in the report. Also include an image, or two, of how you are measuring the length and width of the objects. Example: If you are measuring ivy leaves, you might show a picture of an ivy leaf flat on a table with a ruler indicating what you are considering to be length and width.

Histogram and Boxplot Analysis Using Summary Statistics:

· Using StatKey (or other online stats packages), create a histogram and boxplot of your ratio column. Using One Quantitative Variable tab, you can enter your own data into StatKey under “edit data” or “upload file”. Use screen capture (not a photograph of the screen) to clip the histogram and box plot from StatKey and place them here.

Example: Here is the histogram for an underlying data set.

(This example is not about the golden ratio):

· Describe the shape and symmetry of the histogram and which way (left/right) the mean is skewed away from the median. Next, measure the relative skewing by using the standard deviation (from your data set) as a ruler. Meaning, how many standard deviations fit inside the absolute distance between mean and the median?

Example: Here is the boxplot for an underlying data set.

(This example is not about the golden ratio):

· Describing the variance of the data set: We want to evaluate each quartile for variance. State the width of each quartile in absolute value. Example: Quartile 1 is the lower 25% of the data set. Thus the width of quartile 1 is . The width of quartile 2 would be and so on. Note that is referred to as the Median. You should have four measures, one for the width of each quartile. You do not need to describe the calculations, but instead just least these measures in order and which quartile they represent.

· Describe the relative variance of the middle 50% (IQR).

What do you think this number is describing? What number do you think is a large, or small, amount of variance?

Hypothesis Test, Confidence Interval and Conclusions:

· State the hypotheses.

· Calculate the theoretic Standard Error (S.E.) using your data set (6.2 in text).

· Calculate the t-statistic (think z score).

· Using the theoretic t-distribution in StatKey, find the p-value for your t-statistic (and therefore of the mean for your ratio column).

· Describe in detail in the context of the investigation what the p-value means. (Example using human body temp: If we assume the average human body temperature is , we find the probability of our statistic occurring by random natural variance is …)

· State a conclusion about the hypotheses using a significance level of

· Create a 95% confidence interval (using the SE above) for the true parameter and describe specifically what you are 95% confident about.

Bias and Critiques: In this section you should critique your own work. How well do you think your randomization scheme worked in preventing bias? Describe at least one form of bias in your data set (there will be at least one). What would you do to improve it next time? How accurately do you think your measurements were? What could you do to improve this? Do you believe your conclusions are reliable based on the underlying data set? And anything else you would like to improve or do differently in the future.

Personal Reflections: (there is no right or wrong here, except to leave it blank)

○ What parts of this project were difficult for you? Were there technology issues that got in your way? Did you find the writing center/math tutoring helpful (if used)?

○ What did this project help you understand about these statistical tools?

○ What do you think this project measures about you and your understanding of the material and in what way is this measurement different from a timed exam (is this more work)?

· Do you prefer timed exams?

○ How could this project be improved?

Data Set: Embed a table containing all of your measurements and data.

***Notes to you:

· DO NOT write in bullet point form. Write in the fashion the professional papers we have viewed.

· There is no minimum length. The report should be as long as is needed to discuss all of the above.

· All of my example images above are from StatKey and are NOT about the golden ratio. They are from varied data sets and are there as examples of how to place screen captures in your document.

· DO NOT send your final project to me via email or a Googl.doc link. The final report needs to be a word.doc or a PDF uploaded to Canvas in the link provided for the project. (You can work on this in Google.doc if you wish, but you will need to download as a word.doc when finished and then upload to Canvas link.)

· Feel free to discuss with tutors, fellow classmates and myself. Every project must use a unique data set that is collected by you, thus everyone’s findings will naturally be slightly different.

· Success on this project is not about finding the golden ratio. Success is about analyzing your results and meaningfully describing them with relevant statistics.

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