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Sometimes two different data sets can be collected from the same source. Graphing these data sets in a scatter plot and fitting a mathematical model to the data can be a helpful analysis tool.

A scatter plot is a graph that relates numerical data with two parameters in a coordinate plane. The data points are plotted as ordered pairs. For instance, the number of ice cream cones sold daily at a kiosk can be plotted against the temperature that day.

Each point shows the temperature and ice cream cone sales for a particular day. Notice how, since it's more likely that ice cream salesWhen there is a statistical connection between two parameters of data, such that a change in one is associated with a change in the other, they are said to be correlated. For instance, up until approximately age $18,$ there is a correlation between age and height: older people are generally taller, and taller people are generally older.

Causation is a relationship between two correlated quantities where one **directly** affects the other.

**Causation exists**: An example of a correlation where there is also a causation is height and age. Aging directly causes growth, up until some point.

**No causation**: In winter, both the number of house fires and car accidents increase — they are correlated. However, house fires do not cause car accidents. There is a potential common factor that can explain the increase in both: winter, which causes both slippery road conditions and more candles to be lit, which leads to more fires. Here, there is a correlation, but no causation.

If two quantities correlate in such a way that an increase of one quantity is associated with an increase in the other, they are said to be *positively correlated*. Likewise, an increase in one quantity associated with a decrease in the other, is called a *negative correlation*.

The more the data points appear to follow a specific trend, the more correlated they are. If they are situated almost exactly on a line, the quantities are said to be *strongly correlated*, while if they are more spread out, the quantities are *weakly correlated*.

How strongly two quantities correlate can be described using the correlation coefficient, $r.$ It can take on values between $\text{-}1$ and $1.$ Values near $\text{-}1$ means that the correlation is strong and negative, while a strong, positive correlation leads to a value close to $1.$ If there is no correlation, it has a value of $0.$

Data that has been drawn in a scatter plot and shows a moderate or strong correlation can be modeled using a line of fit. This is a line drawn on the scatter plot that is as close to as many data points as possible. For instance, a line of fit can be drawn for the weight of kittens plotted against their age.

A line of fit can be used to make predictions and generalize the trends of data sets. When a line of fit is determined using strict mathematical methods, it is commonly referred to as a line of best fit.

The music-and-animal amusement park "Hiphop-opotamus Park" has $20$ hippos that they regularly measure. The length and height of each hippo at the time of the last measurement has been plotted below.

Draw a line of fit to the data. Then, comment on the apparent association in the data. Lastly, estimate the equation of the line and interpret its slope and $y$-intercept in context.

A line of fit should be drawn so that as many points as possible are close to the line, with roughly half above and half below.

Looking at the graph, we can see that an increase in height is associated with an increase in length. Thus, the line of fit has a positive slope. Therefore, there is a positive correlation between a hippo's height and length. Note that height does not directly cause length, nor does length directly cause height. Thus, we can argue that there is no causation between these quantities, only a correlation.

Now that a line of fit has been drawn, we can approximate its equation in point-slope form. To find the slope of the line, we'll use two points that are on the line — not necessarily points from the data set.

It can be seen that $(1,5)$ and $(4.5,10)$ lie on the line. We'll substitute these points into the slope formula.$m=\dfrac{y_2-y_1}{x_2-x_1}$

$m=\dfrac{{\color{#009600}{10}}-{\color{#0000FF}{5}}}{{\color{#009600}{4.5}}-{\color{#0000FF}{1}}}$

$m=\dfrac{5}{3.5}$

$m\approx1.43$

The calculations above approximate the following about the line of fit. $m=\frac{1.43}{1} \quad \text{and} \quad b=3.57.$
Since $x$ represents the height of a hippo and $y$ represents the length, the slope tells us that for every $1$ foot in height the hippo has an additional $1.43$ feet in length.

Similarly, the $y$-intercept means that a hippo whose height measures $0$ feet will be $3.57$ feet long. Notice that this last statement doesn't make sense. This is because the point $(0,3.57)$ lies **outside** the given data set. The relationship established for the given data set only applies to values that fall within it. In other words, extrapolation is not reliable while interpolation is.

When a line of fit has been drawn on a scatter plot, it is possible to determine how well the line models the data. This can be done by analyzing the residuals. A residual is the difference between a data point and the line of fit.

Generally, the smaller the absolute values of the residuals, the more reliable the line of fit is. If the residuals are graphed in a scatter plot, a random or non-uniform distribution indicates a reliable line of fit. Likewise, if some kind of pattern appears in the scatter plot, the line is probably not a good fit for the data.Most graphing calculators have a function called *linear regression* which can be used to find a precise line of fit, using strict rules. This line of fit is then called the line of best fit or the regression line.

The first step is to enter the data points in the calculator. On a TI calculator, this is done by first pressing the STAT button, and then selecting the option Edit....

This gives a number of columns, marked L$1$, L$2$, L$3,$ etc.

With the help of the arrow keys, choose where in the lists to fill in the data values. Enter the data points' $x$-values in list L$_1,$ and the corresponding $y$-values in L$_2.$ Values are entered into the fields using the number buttons followed by ENTER.

After entering the values, press the STAT button, then select the menu item CALC to the right.

The option LinReg(ax+b) gives a line of best fit, expressed as a linear function in slope-intercept form.

In this case, the line of best fit is described by the function $y= \text{-} 5.92x + 28.8.$ The correlation coefficient, $r,$ is roughly $\text{-} 0.99,$ indicating a strong negative correlation. If $r$ doesn't show up, press CATALOG (2ND + $0$), scroll down to the option DiagnosticOn and enable it by pressing ENTER twice. Then, find the line of best fit again.

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