In this question,
we're asked to use linear regression to find the equation
for the linear function that best fits the given data
and we're told to round to two decimal places
and write the equation in slope intercept form
where the form y equal mx plus b.
We will also find r, the correlation coefficient,
as well as r-squared, the coefficient of determination,
rounded to four decimal places.
Using a TI-84, the first step is to turn
the diagnostic tool on so that we are given r and r-squared.
To do this, we need to go to the catalog
and select diagnostic on.
We press second zero for the catalog
and we can scroll down from here to diagnostic on
where we can press the D button,
which is this button here,
which brings us down to the D's
and then we can scroll down from here.
Again, we are looking for diagnostic on,
which is here.
So we press enter
and then enter again.
Now when we perform the linear regression,
the calculator will give us r and r-squared.
If your calculator does not show r and r-squared,
your diagnostic tool is not turned on.
And now we enter the data by pressing stat and then enter.
Notice how there's some old data in L one.
To clear this, we go to the top of the column,
press clear and then enter.
And now we enter the x values in L one
and the y values in L two.
Right arrow to L two
and now enter the y values.
It's worth double-checking these values
because if one value is off,
we will not get the correct regression equation,
and now we go back to the home screen
by pressing second mode for quit
and perform linear regression, we press stat,
right arrow once to calculate,
option four for linear regression.
On this screen
because we have the x values in L one
and the y values in L two,
we don't change the first two rows,
in the third row,
we are not using a frequency list,
we leave this blank,
in the fourth row,
we could store the regression equation in,
for example, y one,
but we don't need to for this question.
We simply go down to calculate and press enter.
And now we have all the information we need.
Let's first identify the regression equation
in the form of y equals ax plus b
or a is the slope, which is equal to 24
and b, the constant, which is the vertical intercept,
rounded to two decimal places
is approximately 56.67,
which means our regression equation is
y equals 24x plus 56.67,
and now let's find r and r-squared,
rounded to four decimal places.
Notice r is the value on the bottom.
r to four decimal places is 0.9978
and r-squared to four decimal places is 0.9955.
So let's go ahead and write these down
and then talk about what these mean.
Let's first talk about the correlation coefficient,
often referred to as r.
The value of r is always an interval
from negative one to positive one.
The sign of r indicates the direction
of the linear relationship between the two variables.
If r is less than zero or negative,
there is a negative linear relationship
between the two variables and therefore
the slope of the regression equation is negative.
If r is greater than zero or positive, like in our case,
there is a positive linear relationship
between the two variables
and the slope of the regression equation is positive,
which we can verify from our results.
Notice how r is positive and the slope
of the regression equation is also positive.
The magnitude of r indicates the strength
of the linear relationship between the two variables.
If r is equal to zero,
there is no linear relationship between the two variables.
If the absolute value of r is approximately one,
then there is a strong linear relationship
between the two variables
and if r is equal to one or equal to negative one,
there is an exact linear relationship
between the two variables,
which means every data point falls exactly
on the regression line.
And now let's talk about
what the coefficient of determination indicates.
The coefficient of determination of r-squared
is always an interval from zero to one
and it measures how close the data values
are fitted to the regression line.
It measures the proportion of the variance
in the dependent variable or the output, which is y,
that is predictable from the independent variable
or the input, which is x,
and again, r-squared is always an interval from zero to one.
I hope you found this helpful.