A. use technology to create a scatterplot, to determine the...

a. use technology to create a scatterplot, to determine the best fit line, and to compute $r$ b. indicate whether or not the best fit line can be used for predictive purposes for the following $x$ -values. For those for which it can be used, give the predicted value of $y$. i. $x=0$ ii. $x=-6$ iii. $x=12$ iv. $x=-15$ c. Using the best fit line, at what $x$ -value would you expect $y$ to be equal to $2 ?$ $$\begin{array}{|c|c|} \hline x & y \\ \hline-20 & 15 \\ \hline-18 & 10 \\ \hline-14 & 3 \\ \hline-14 & 8 \\ \hline-13 & 3 \\ \hline-8 & 0 \\ \hline-8 & -3 \\ \hline-5 & -6 \\ \hline 1 & -11 \\ \hline 1 & -15 \\ \hline \end{array}$$

a. use technology to create a scatterplot, to determine the best fit line, and to compute $r$
b. indicate whether or not the best fit line can be used for predictive purposes for the following $x$ -values. For those for which it can be used, give the predicted value of $y$.
i. $x=0$
ii. $x=-6$
iii. $x=12$
iv. $x=-15$
c. Using the best fit line, at what $x$ -value would you expect $y$
to be equal to $2 ?$
$$\begin{array}{|c|c|}
\hline x & y \\
\hline-20 & 15 \\
\hline-18 & 10 \\
\hline-14 & 3 \\
\hline-14 & 8 \\
\hline-13 & 3 \\
\hline-8 & 0 \\
\hline-8 & -3 \\
\hline-5 & -6 \\
\hline 1 & -11 \\
\hline 1 & -15 \\
\hline
\end{array}$$

Key Concepts

Extrapolation vs. Interpolation

Interpolation refers to making predictions within the range of the observed data values, which tends to be more reliable since it is based on the established relationship. Extrapolation, on the other hand, involves making predictions outside of the range of observed data, which can be less reliable due to the possibility that the underlying relationship may not hold outside the observed span.

Correlation Coefficient (r)

The correlation coefficient, denoted by r, is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. Its value ranges from -1 to 1, where values close to 1 or -1 indicate a strong relationship, and values near 0 suggest a weak linear association.

Prediction Using the Regression Equation

Using the regression equation derived from linear regression allows one to predict the value of the dependent variable for any given value of the independent variable. However, the reliability of these predictions depends on whether the input value lies within the range of observed data.

Scatterplot

A scatterplot is a graphical representation of two quantitative variables that uses dots to display the values obtained for each variable. It helps in visually assessing the relationship between the variables, identifying patterns, and detecting potential outliers or anomalies in the data set.

Linear Regression (Best Fit Line)

Linear regression involves finding the best-fitting straight line through a set of data points, usually determined by the least squares method. This best fit line minimizes the sum of the squared differences between observed values and the values predicted by the line, providing a simple model to describe the linear relationship between variables.

Transcript

00:03 We are given a data set of x and y coordinate pairs, and we're tasked with using a graphing utility to graph the data and determine some things about the correlation.

00:15 So first off, this was done using excel, and excel not only made us this nice scatterplot, but it also, using the correlation function, calculated for us that the r value is negative 0 .98022.

00:30 Furthermore, we were able to determine our equation for the line of best fit.

00:36 And so we have what we need here.

00:39 The next task we had was to figure out what the value of the y would be if x is zero, negative 6, 12, or negative 15.

00:53 And of course, we can only do that if the data is reasonably predictive for that x value.

00:59 So if x is zero, that would put us right about there at negative 12.

01:05 And since zero is within the range of x values of our actual data, and our lowest x value was negative 20, and our highest x value was one.

01:17 So anything in between there, it's clearly okay, and anything beyond needs to be very, very close.

01:24 So f of zero is what we get when we substitute zero into, our equation here.

01:31 So this one's pretty straightforward.

01:33 You don't need a calculator for it.

01:35 Looking at the graph, we can see it's about negative 12.

01:38 And in fact, our slope times zero is zero.

01:41 And all that's left is the y intercept, which is negative 11 .9 .9.

01:53 And our next one, negative six, negative six is inside that range of x's.

01:59 So negative six is right about there.

02:02 And that would put us, but halfway between negative 4 and negative 5.

02:07 So if we don't fumble with the calculator, we should get something in the neighborhood of negative 4 .5.

02:12 So let's go ahead and calculate that one.

02:15 So f of negative 6 is going to equal negative 1 .2 -631 times negative 6 minus 11 .97.

02:38 And the calculator says that equals negative 4 .404.

02:49 So that's pretty close to negative 4 .5.

02:52 We'll assume we did not fumble on the calculator.

02:55 F of 12, 12 is so much larger than 2, which is the highest x value we had for our actual data...

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