Project 4

Overview. In this project, you will use linear regression to analyze the relationship between the length of a person's metacarpal bone and the person's height.

Purpose. To test your understanding of elementary linear regression.

Project guidelines. Remember that the projects in this course are in lieu of exams, so you are not to discuss them or receive any input or help from anyone else on them. If you have any questions about a project, you should ask me.

Your write-up should be in prose (and not as an itemized list or, worse yet, a mere sequence of equations) and should be directed toward a reader who has no familiarity whatsoever with the study that you conducted. As such, you should include information regarding how and why it was conducted (including the main question of interest) as well as all information necessary for the reader to understand, assess, and interpret your results.

As with the previous projects, write your paper to present the entire study to someone who is familiar with statistics at the level you are but who is not familiar with your project. You should include in the introduction what the primary question of interest is and all relevant information about the process that you use to investigate this question. Because you don't have direct access to the original data source, however, you do not need to critique the process in this project.

The procedure. In anthropology and forensic investigations, scientists sometimes want to be able to estimate the height of a human being from a small bone that has been found, such as metacarpal bone I, which is a bone in the hand that connects to the thumb. For this project, the main question of interest is: What is the relationship between the length of the metacarpal bone I and a person's height?

For this project, you may want to use the notation that $M$ is the length of a person's metacarpal bone in inches and $H$ is a person's height. If you do, you should explain in your paper that you will use this notation.

In this project, you should postulate that there is a linear relationship between $M$ and $H$, meaning that:

\(\mu(H \mid M)=a+b M .\)

You will fit the coefficients \(a\) and \(b\) to the data set given in the course website.

Ordinarily if you were conducting an analysis of such data, you should consult the original data source to learn as much as possible about the context of the data, but you do not need to do so here. (And actually you should not do so here, since you should not compare your answers to any of theirs, which are somewhat different anyway.) You should, however, cite this paper as the source of the data.

The data set contains the length of metacarpal bone I and the height of 9 adult human beings, both variables being given in inches.

In your paper, be sure to do at least the following things (with the section that each item should be in given in parentheses):

1. (Introduction) State what the purpose of the paper is, how you will accomplish this purpose, and what is contained in each section of the paper.
2. (The data) State where the data came from and display the data in a table (including units in the column headings). If you are going to use any abbreviations for the variables, explain those here as well.
3. (The data) Display a scatterplot of height versus metacarpal bone I length without a regression line. You should make this plot in Excel and then cut and paste it into your paper. Be sure to label the axes (including units).
4. (The data) Using the scatterplot, state whether or not linear regression appears appropriate. If it is, proceed; if not, state clearly that it is not, but proceed anyway. As part of this, state what you can tell from the scatterplot about the form, direction, and strength of association between the two variables.
5. (Analysis) Display a scatterplot of height versus metacarpal bone I length with a regression line. You should make this plot in Excel and then cut and paste it into your paper. Be sure to label the axes (including units). You may also include the regression line equation and correlation squared in the plot if you like. (This isn't necessary, but you will need that information later in your paper.)
6. (Analysis) Give the regression equation of height modeled on metacarpal bone I length. This should be of the form:
   \(\mu(H \mid M)=a+b M,\)
   using the regression coefficients \(a\) and \(b\) that you have used Excel to find. Be sure to round your values for \(a\) and \(b\) suitably and to include units for \(a\) and \(b\) in this equation.
7. (Analysis) Give the correlation squared of \(M\) and \(H\), rounded suitably. (Correlation is unitless, however.)
8. (Conclusions) Restate the regression equation, noting that this answers your main question of interest.
9. (Conclusions) To show how the regression equation can be used, use it to estimate the mean height of adult human beings who have a metacarpal bone I of length $2.1$ inches.
10. (Conclusions) Interpret the regression coefficient \(a\) in terms of what it tells you about metacarpal bone I length and height. Explain why, although this interpretation first appears strange, there really isn't anything strange about it.
11. (Conclusions) Interpret the regression coefficient \(b\) in terms of what it tells you about metacarpal bone I length and height.
12. (Conclusions) Interpret the correlation squared of \(M\) and \(H\).

Price: $11.06

Solution: The downloadable solution consists of 7 pages, 406 words and 6 charts.
Deliverable: Word Document