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Section 3.3 Eyeballing Lines of Best Fit

Subsection 3.3.1 Overview

Sometimes data is not perfectly linear, but has a linear trend. An example follows.

Student Page 3.3.2 Pricing Canisters
A4US

The height and price of each of the canisters in Figure 3.3.2.1 is provided in Table 3.3.2.2:

Figure 3.3.2.1. Canisters
Table 3.3.2.2. Height and Price of Canisters
Height (in inches) Price (in dollars)
7 24.95
8.75 29.95
12 39.95

1.

Would you expect the data in the table to be linear? Why or why not?

2.

Is the relationship between the height and the price of the canisters linear? Why or why not?

3.

Is the data perfectly linear? If not, which canister seems to be priced too high or too low? Explain your thinking.

In this lesson, we will consider other experiments and data sets. We will first determine if we think a linear trend would be a good fit. If so, we will eyeball a line of best fit, determine an equation for the eyeballed line, and use the line to test data and make predictions.

A line of best fit has these properties:

List 3.3.2.3.

  1. The direction of the line is evident from the data.

  2. As many points as possible should be on the line as long as the same number of points are above and below it.

  3. The points above the line should not be concentrated at one end of the line nor should the points below the line.

Student Page 3.3.3 Gulliver Graphs
A4US

In Gulliver’s Travels, the Lilliputians made a coat for Gulliver that fit him perfectly. They did this after taking only one measurement, the circumference of Gulliver's thumb. The Lilliputians then used the following well-known Lilliputian “Rule of Thumb”:

The circumferences of each body part are related as in the following equations:

\(\text{Wrist} = 2 \cdot \text{Thumb}\text{,}\) \(\text{Neck} = 2 \cdot \text{Wrist}\text{,}\) and \(\text{Waist} = 2 \cdot \text{Neck}\)

1.

By gender, gather thumb, wrist, and neck measurements, in centimeters, for at least 20 adults. Measure the thumb between the knuckles. Measure the wrist on the arm just above the wrist bones. Measure the neck just below the chin. Put your data in Table 3.3.3.1.

2.

Graph the data for (Thumb, Wrist), one graph each for females, males, and entire data set. Use a different color for each data set.

3.

Draw lines that best fit each data set. Explain why you think each line is a reasonable fit.

4.

Test the Lilliputian “Rule of Thumb” for:
(a)

The female sample

(b)

The male sample

(c)

The entire sample

6.

Were the Lilliputians lucky that Gulliver's coat fit or were they quite clever? Explain.

Table 3.3.3.1. Thumb, Wrist, and Neck Data
Female Male
Thumb Wrist Neck Thumb Wrist Neck
           
           
           
           
           
           
           
           
           
           
           

Functions for Females:

\begin{align*} W(T) \amp = \fillinmath{XXXXXXXXXX}\\ N(W) \amp = \fillinmath{XXXXXXXXXX}\\ N(T) \amp = \fillinmath{XXXXXXXXXX} \end{align*}

Functions for Males:

\begin{align*} W(T) \amp = \fillinmath{XXXXXXXXXX}\\ N(W) \amp = \fillinmath{XXXXXXXXXX}\\ N(T) \amp = \fillinmath{XXXXXXXXXX} \end{align*}

Functions for Entire Sample:

\begin{align*} W(T) \amp = \fillinmath{XXXXXXXXXX}\\ N(W) \amp = \fillinmath{XXXXXXXXXX}\\ N(T) \amp = \fillinmath{XXXXXXXXXX} \end{align*}

A Word about Function Notation.

When you were asked to record the functions you found for Gulliver Graphs, did you find the notation a bit strange? Notice that the notation gives important information about the columns of data being related to each other. \(W(T)\) indicates that we want to know the wrist measurement in terms of the thumb measurement, instead of just using \(W\text{.}\) We read the symbols, \(W\) of \(T\text{.}\) What do you think \(N(W)\) represents? \(N(T)\text{?}\)

If we want to be very specific, we could indicate \(W_F(T_F)\) to indicate the that we want to know the wrist measurements for females in terms of the thumb measurements for females. How would we write the wrist measurement for males in terms of the thumb measurements for males? We are not using subscript notation in Gulliver Graphs, instead opting to use headings to make clear which data sets are being related.

From this point on, we will use function notation to avoid ambiguity in cases where it is necessary to make clear which quantities we are relating.

Homework 3.3.4 Homework

1.

Table 3.3.4.1 shows U.S. shoe sizes for women as compared with foot lengths in inches and centimeters and Euro and UK shoe sizes (Source 38 , retrieved April 22, 2020). Let U.S. shoe sizes be the independent variable. Choose a variable to represent each column of data. Use function notation when recording equations.

Table 3.3.4.1. Shoe Sizes for Women
U.S. Sizes Inches CM Euro Sizes UK Sizes
4 8.1875 20.8 35 2
4.5 8.375 21.3 35 2.5
5 8.5 21.6 35-36 3
5.5 8.75 22.2 36 3.5
6 8.875 22.5 36-37 4
6.5 9.0625 23 37 4.5
7 9.25 23.5 37-38 5
7.5 9.375 23.8 38 5.5
8 9.5 24.1 38-39 6
8.5 9.6875 24.6 39 6.5
9 9.875 25.1 39-40 7
9.5 10 25.4 40 7.5
10 10.1875 25.9 40-41 8
10.5 10.3125 26.2 41 8.5
11 10.5 26.7 41-42 9
11.5 10.6875 27.1 42 9.5
12 10.1875 27.6 42-43 10
(a)

Graph the data, (US sizes, length measurement) for inches or centimeters. Is the data linear? Why or why not? Estimate a line of best fit. Find an equation for the line.

(b)

Graph the data, (US sizes, Other sizes) for Euro or UK. If you choose Euro Sizes, what should you do to accommodate the range of sizes shown for US half sizes? Is the data linear? Why or why not? Estimate a line of best fit. Find an equation for the line.

(c)

Are any of the data sets perfectly linear? If so, which ones? If not, what do you think causes the data set to not be perfectly linear?

2.

Have you played Monopoly? The graph in Figure 3.3.4.2 shows the number of the space (after GO) and the price for the space on a Monopoly game board. GO is space 0. (You can look up the gameboard on the Internet.)

Figure 3.3.4.2. Monopoly: Space Number vs. Price
(a)

Does a linear model appear to fit the data? Explain your thinking.

(b)

On the graph in Figure 3.3.4.2, draw your best guess for a line of best fit for the data (see List 3.3.2.3 for a reminder for properties of a line of best fit).

(c)

Find an equation for your hand-drawn line. Show your work!

(d)

What does the slope mean in terms of Monopoly space numbers and prices?

(e)

What does the \(y\)-intercept mean in terms of Monopoly space numbers and prices?

(f)

What is the independent variable? Why do you think so?

3.

The domain of a function is the set of all possible values of the independent variable (\(x\) or whatever variable you are using for the context) for which the function is defined. The range is the set of possible values of the dependent variable (usually \(y\text{,}\) sometimes expressed as \(f(x)\)) that result from using the same function.

(a)

What is the domain of the function in the graph in Exercise 3.3.4.2 in terms of the context?

(b)

What is the range of the function in the graph in Exercise 3.3.4.2 in terms of the context?

zappos.com/c/shoe-size-conversion