unknown length containing $x$ and $y$ values and then computes a line that best fits the data set. Remember, a

line is defined by the equation $y=mx+b,$ where $m$ is the slope and $b$ is the $y-$intercept of the line. Such a

linear fit is commonly used to visualize linear trends of data. You will then plot the data set using a scatter

plot and a line representing your linear regression of the data. An example of this is shown by Figure $1,$

where the blue dots are the $(x,y)$ points from a data set, and the black line is a linear fit to the data.

Figure $1.$ Example of a linear fit $($black line$)$ of a data set $($blue dots$)$ on an $x-y$ plot.

Algorithm:

Recall, the slope of a line, $m,$ can be computed using any two points on the line, $(x_1,y_1)$ and $(x_2,y_2)$

using the ratio of the "change in $y"$ divided by the "change in $x".$ That is:

$m=\frac{y_2-y_1}{x_2-x_1}$

Once the slope is known, the $y-$intercept can also be computed using the formula: $b=m{x}_1-y_1.$

For the data set such as the blue points in Figure $1,$ we are not fitting a line between any two points, since

no two points are guaranteed to be on the line. Rather, a line is constructed in an average sense. The

following explains the algorithm on how to do this.

Let $\stackrel{}{x}$ be the array of $x-$values in the data set, and $\frac{?}{\text{b}\text{a}\text{r}}(y)$ be the array of $y-$values of the data set, where one of

the blue data point on the graph in Figure $1$ is $).$ Next, let mean $x$ represent the mean value $($or

the average value$)$ of the array $\stackrel{}{x}$ and mean $y$ be the mean $($or average$)$ value of the array $\frac{?}{\text{b}\text{a}\text{r}}(y).$ The

average value of $\stackrel{}{x}$ can be written as an equation:

mean $-x=\frac{1}{N}{\displaystyle \underset{i=0}{\overset{N-1}{}}}x[i]$

where $x[i]$ are the array values of $\stackrel{}{x},$ and ${\displaystyle \underset{i=0}{\overset{N-1}{}}}x[i]$ is the sum of all the array values. A similar expression

can be written for mean $y.$

After computing mean $x$ and mean $y,$ the slop of the fit line can be computed. This is done in an average

sense using the ratio of the sum of the difference of all $y$ values from the $y-$mean, times the difference of

all the $x-$values from the $x-$mean, divided by the difference of all the $x-$values from the $x-$mean. This can

be written as an equation as:

$)-)-)-)-$

where the numerator and denominator are separate sums.

Given the slope, the y$-$intercept is simply:

$)-)-$

Using this algorithm, each point on the black line in Figure $1$ is simply computed as:

$y_{\text{f}\text{i}\text{t}}[j]=m_{\text{f}\text{i}\text{t}}^{**}x[j]+b_{\text{f}\text{i}\text{t}}.$

Functions needed for this project:

As part of this C$++$ project, you will need to develop at least five functions.

The first function is to read in a data set of unknown length that contains two columns of

data. You should store this in a single $($one$-$dimensional$)$ array that stores the two values

by column. That is$,$ if there are $N$ data points, the array will be of length $2N.$ The first $N$

values will be your $x-$array, and the second $N-$values will be your $y-$array. The function

should be passed the name of a file it will read in the $(x,y)$ values from, and the function

will dynamically create an array to store the $(x,y)$ values as just described, and then return

a pointer to the array and the length of the array.

The second function computes and returns the average $($or mean$)$ value of an array of

some length $N.$

The third function is passed the array storing the