## Learning Math: Data Analysis, Statistics, and Probability

# Bivariate Data and Analysis Homework

Is there an association between the length of your forearm (elbow to wrist) and the length of your feet?

The following are measurements, in millimeters, for forearm length and right-foot length for 20 people:

N = 20 measurements

Mean forearm length = 262.1

Mean foot length = 255.7

**Scatter Plot**

**Problem H1
**Describe the association between forearm length and foot length.

**Problem H2
**Use the mean forearm length and the mean foot length to determine the quadrants and number of points in each quadrant.

**Problem H3
a. **Use the quadrants from Problem H2 to create the contingency table for this data.

**What percentage of people with above-average forearm lengths also have above-average foot lengths?**

b.

b.

**What percentage of people with above-average forearm lengths also have below-average foot lengths?**

c.

c.

**What percentage of people with below-average forearm lengths also have below-average foot lengths?**

d.

d.

**What percentage of people with below-average forearm lengths also have above-average foot lengths?**

e.

e.

**What do these percentages say about the strength of the association between forearm length and foot length?**

f.

f.

**Problem H4**

Consider the line Foot Length = Forearm Length (YL = X).

**a.** Complete the table below.

**b.** Determine the SSE for this data.

When you click “Show Answers,” the filled-in table will appear below the problem. Scroll down the page to see it.

**Problem H5
**Consider the line Foot Length = Forearm Length + 4 (YL = X + 4).

**a.**Complete the table below.

**b.**Determine the SSE for this data.

**Problem H6
**Compare the SSE in Problem H4 with the SSE in Problem H5. Which line provides a “better” fit? Explain.

### Solutions

**Problem H1**

Overall, there is a positive association between forearm length and foot length. On the graph, the points generally go up and to the right.

**Problem H2
**To find the quadrants, we must use the mean forearm and foot lengths, which we know are 262.1 mm and 255.7 mm, respectively.

Recall that:

• |
Quadrant I has points that correspond to people with above-average forearm and foot lengths. |

• |
Quadrant II has points that correspond to people with below-average forearm lengths and above-average foot lengths. |

• |
Quadrant III has points that correspond to people with below-average forearm and foot lengths. |

• |
Quadrant IV has points that correspond to people with above-average forearm lengths and below-average foot lengths. |

Here is the scatter plot divided into quadrants:

This table shows which quadrant each point is in:

**Problem H3**

a. |
The contingency table is above. |

b. |
Of the eight people with above-average forearm lengths, 87.5% (7 / 8) also have above-average foot lengths. |

c. |
Of the eight people with above-average forearm lengths, only 12.5% (1/ 8) have below-average foot lengths. |

d. |
Of the 12 people with below-average forearm lengths, 83.3% (10 / 12) also have below-average foot lengths. |

e. |
Of the 11 people with below-average forearm lengths, only 16.7% (2 / 12) have above-average foot lengths. |

f. |
These percentages say that there is a fairly strong (more than 80%) positive association between forearm length and foot length. |

**Problem H4:
**

**a.**Here is the completed table:

**b.** The SSE, (256 + 324 + … + 400 + 1), is 3,374.

**Problem H5
a.** Here is the completed table:

**b. **The SSE, (400 + 196 +… + 576 + 25), is 4,558.

**Problem H6**

The first SSE is smaller, which means that the line Foot Length = Forearm Length is a better fit to the data than the line Foot Length = Forearm Length + 4. Here is an illustration of these two lines on top of the data set:

As you can see from the graph, the line Foot Length = Forearm Length is a closer representation of the data than the line Foot Length = Forearm Length + 4.