You might be familiar with calorie requirements for males, like the ones shown in the table below. What type of correlation is exhibited by the data?

**Calorie Requirements (Male), 1-59 years**

Variable | Values | |||||
---|---|---|---|---|---|---|

Age Range, \begin{align*}x\end{align*} |
1-3 | 4-6 | 7-10 | 11-14 | 15-18 | 19-59 |

Calorie Needs, \begin{align*}y\end{align*} |
1230 | 1715 | 1970 | 2220 | 2755 | 2550 |

The age is measured in years.

### Scatterplots and Equation of Best Fit

A **scatterplot** is a set of points that represent data. We plot these points and try to find equations that best approximate their relationship. Because data is not always perfect, not every point will always fit on the line of best fit. The **line of best fit** is the line that is the closest to all the data points. It can be used to approximate data within the set or beyond the set. Scatterplots almost always represent a real-life situation.

Scatterplots can have **positive correlation** if the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values tend to increase together. They can have **negative correlation** if \begin{align*}y\end{align*} tends to decrease as \begin{align*}x\end{align*} tends to increase. And, if the points have no sort of linear pattern, then the data would have relatively **no correlation.** Think of the type of correlations referring to the slope of the line that would best fit that data.

Let's describe the type of correlation shown in the scatterplot and explain the answer.

Source: CNN

This is a negative correlation. As the years get larger, the sales go down. This could be because in the boom of online/digital and pirated music.

Now, let's find the linear equation of best fit for the data set above.

First, it can be very difficult to determine the “best” equation for a set of points. In general, you can use these steps to help you.

Step 1: Draw the scatterplot on a graph.

Step 2: Sketch the line that appears to most closely follow the data. Try to have the same number of points above and below the line.

Step 3: Choose two points on the line and estimate their coordinates. These points do not have to be part of the original data set.

Step 4: Find the equation of the line that passes through the two points from Step 3.

Let’s use these steps on the graph above. We already have the scatterplot drawn, so let’s sketch a couple lines to find the one that best fits the data.

From the lines in the graph, it looks like the purple line might be the best choice. The red line looks good from 2006-2009, but in the beginning, all the data is above it. The green line is well below all the early data as well. Only the purple line cuts through the first few data points, and then splits the last few years. Remember, it is very important to have the same number of points above and below the line.

Using the purple line, we need to find two points on it. The second point, crosses the grid perfectly at (2000, 14). Be careful! Our graph starts at 1999, so that would be considered zero. Therefore, (2000, 14) is actually (1, 14). The line also crosses perfectly at (2007, 10) or (8, 10). Now, let’s find the slope and \begin{align*}y-\end{align*}intercept.

\begin{align*}m = \frac{14-10}{1-8} = - \frac{4}{7}\end{align*}

\begin{align*}y &= - \frac{4}{7}x+b\\ 14 &= - \frac{4}{7}(1)+b \\ 14 &= -0.57+b\\ 14.57 &= b\end{align*}

The equation of best fit is \begin{align*}y = - \frac{4}{7}x+14.57\end{align*}.

However, the equation above assumes that \begin{align*}x\end{align*} starts with zero. In actuality, we started with 1999, so our final equation is \begin{align*}y = - \frac{4}{7}(x-1999)+14.57\end{align*}.

Using the line of best fit above, what would you expect music sales to be in 2010?

In this problem, we are using the line of best fit to predict data. Plug in 2010 for \begin{align*}x\end{align*} and solve for \begin{align*}y\end{align*}.

\begin{align*}y &= - \frac{4}{7}(2010-1999)+14.57\\ y &= - \frac{4}{7}(11)+14.57\\ y &=8.3\end{align*}

It is estimated that music industry will make $8.3 billion in music sales in 2010.

### Examples

#### Example 1

Earlier, you were asked to find the type of correlation that is exhibited by the data.

If you draw a scatter plot of the data, you see that the *x* and *y* values tend to increase together. Therefore the data exhibits positive correlation. That is, as age increases so do calorie requirements.

**Use the table below to answer the following questions.**

Sleep Requirements, 0-3 years

Age, \begin{align*}x\end{align*} |
1 | 3 | 6 | 9 | 12 | 18 | 24 | 36 |
---|---|---|---|---|---|---|---|---|

Sleep, \begin{align*}y\end{align*} |
16 | 15 | 14.25 | 14 | 13.75 | 13.5 | 13 | 12 |

The age is measured in months and sleep is measured in hours.

Source: BabyCenter

Draw a scatterplot with age across the \begin{align*}x-\end{align*}axis and sleep along the \begin{align*}y-\end{align*}axis. Count by 3’s for the \begin{align*}x-\end{align*}values and by 2’s for the \begin{align*}y-\end{align*}values.

Here is the scatterplot.

#### Example 3

Find the line of best fit.

Two points that seem to be on the red line are (3, 15) and (24, 13).

\begin{align*}m &= \frac{15-13}{3-24} = - \frac{2}{21}\\ 15 &= - \frac{2}{21}(3)+b\\ 15 &= -0.29+b\\ 15.29 &= b\end{align*}

The equation of the line is \begin{align*} y = - \frac{2}{21}x+15.29\end{align*}.

#### Example 4

Determine the amount of sleep needed for a \begin{align*}2 \frac{1}{2}\end{align*} year old and a 5 year old.

First, you need to change the age to months so that it corresponds with the units used in the graph. For a 2.5 year-old, 30 months, s/he should sleep \begin{align*}y = - \frac{2}{21}(30)+15.29 \approx 12.4\end{align*} hours. For a 5-year-old, 60 months, s/he should sleep \begin{align*}y = - \frac{2}{21}(60)+15.29 \approx 9.6\end{align*} hours.

### Review

Determine if the scatterplots below have positive, negative, or no correlation.

Plot each scatterplot and then determine the line of best fit.

- .

\begin{align*}x\end{align*} | 1 | 2 | 3 | 5 | 7 | 8 |
---|---|---|---|---|---|---|

\begin{align*}y\end{align*} | 1 | 3 | 4 | 3 | 6 | 7 |

- .

\begin{align*}x\end{align*} | 10 | 9 | 7 | 6 | 5 | 2 |
---|---|---|---|---|---|---|

\begin{align*}y\end{align*} | 5 | 6 | 4 | 3 | 3 | 2 |

Use the data below to answer questions 6-8.

The price of Apple stock from Oct 2009 - Sept 2011 source: Yahoo! Finance

10/09 |
11/09 |
12/09 |
1/10 |
2/10 |
3/10 |
4/10 |
5/10 |
6/10 |
7/10 |
8/10 |
9/10 |
---|---|---|---|---|---|---|---|---|---|---|---|

$181 | $189 | $198 | $214 | $195 | $208 | $236 | $249 | $266 | $248 | $261 | $258 |

10/10 |
11/10 |
12/10 |
1/11 |
2/11 |
3/11 |
4/11 |
5/11 |
6/11 |
7/11 |
8/11 |
9/11 |

$282 | $309 | $316 | $331 | $345 | $352 | $344 | $349 | $346 | $349 | $389 | $379 |

- Draw the scatterplot for the table above. Make the \begin{align*}x-\end{align*}axis the month and the \begin{align*}y-\end{align*}axis the price.
- Find the linear equation of best fit.
- According to your equation, what would be the predicted price of the stock in January 2012?

Use the data below to answer questions 9-11.

Total Number of Home Runs Hit in Major League Baseball, 2000-2010 source: Baseball Almanac

2000 |
2001 |
2002 |
2003 |
2004 |
2005 |
2006 |
2007 |
2008 |
2009 |
2010 |
---|---|---|---|---|---|---|---|---|---|---|

5693 | 5458 | 5059 | 5207 | 5451 | 5017 | 5386 | 4957 | 4878 | 4655 | 4613 |

- Draw the scatterplot for the table above. Make the \begin{align*}x-\end{align*}axis the year and the \begin{align*}y-\end{align*}axis the number of home runs.
- Find the linear equation of best fit.
- According to your equation, how many total home runs should be hit in 2011?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 2.14.