Consider the following data set: x 5 6 7 7 8 8 8 8 y

step 1 All right, guys, let's go ahead and start this one. So the first thing we're supposed to do is d

Consider the following data set: x 5 6 7 ...

Consider the following data set: $$\begin{array}{lllllllll} x & 5 & 6 & 7 & 7 & 8 & 8 & 8 & 8 \\ \hline y & 4.2 & 5 & 5.2 & 5.9 & 6 & 6.2 & 6.1 & 6.9 \\ \hline x & 9 & 9 & 10 & 10 & 11 & 11 & 12 & 12 \\ \hline y & 7.2 & 8 & 8.3 & 7.4 & 8.4 & 7.8 & 8.5 & 9.5 \end{array}$$ (a) Draw a scatter diagram with the $x$ -axis starting at 0 and ending at 30 and with the $y$ -axis starting at 0 and ending at 20 (b) Compute the linear correlation coefficient. (c) Now multiply both $x$ and $y$ by 2 (d) Draw a scatter diagram of the new data with the $x$ axis starting at 0 and ending at 30 and with the $y$ -axis starting at 0 and ending at $20 .$ Compare the scatter diagrams. (c) Compute the linear correlation coefficient. (f) Conclude that multiplying each value in the data set does not affect the correlation between the variables. Explain why this is the case.

Consider the following data set:
$$\begin{array}{lllllllll}
x & 5 & 6 & 7 & 7 & 8 & 8 & 8 & 8 \\
\hline y & 4.2 & 5 & 5.2 & 5.9 & 6 & 6.2 & 6.1 & 6.9 \\
\hline x & 9 & 9 & 10 & 10 & 11 & 11 & 12 & 12 \\
\hline y & 7.2 & 8 & 8.3 & 7.4 & 8.4 & 7.8 & 8.5 & 9.5
\end{array}$$
(a) Draw a scatter diagram with the $x$ -axis starting at 0 and ending at 30 and with the $y$ -axis starting at 0 and ending at 20
(b) Compute the linear correlation coefficient.
(c) Now multiply both $x$ and $y$ by 2
(d) Draw a scatter diagram of the new data with the $x$ axis starting at 0 and ending at 30 and with the $y$ -axis starting at 0 and ending at $20 .$ Compare the scatter diagrams.
(c) Compute the linear correlation coefficient.
(f) Conclude that multiplying each value in the data set does not affect the correlation between the variables. Explain why this is the case.

Key Concepts

Scaling of Data

Scaling refers to the multiplication of data values by a constant factor. This operation changes the numerical values of the data points but does not affect the overall pattern or the strength of the linear relationship between the variables when the transformation is applied uniformly.

Invariance of Correlation Under Linear Transformations

The correlation coefficient remains unchanged under linear transformations, such as multiplication by a constant. This invariance occurs because the transformation affects both the numerator and the denominator of the correlation formula proportionally, leaving the correlation unchanged. Consequently, the scatter diagram's shape remains similar, regardless of uniform scaling.

Scatter Diagram

A scatter diagram is a graphical representation of two variables plotted on a Cartesian coordinate system. It helps in visualizing the relationship between the variables by displaying individual data points, revealing patterns, trends, or potential outliers in the data set.

Linear Correlation Coefficient

The linear correlation coefficient is a statistical measure that quantifies the degree and direction of the linear relationship between two quantitative variables. Its value ranges from -1 to 1, with values close to either extreme indicating a strong linear relationship and values near zero indicating a weak linear relationship.

Transcript

00:01 All right, guys, let's go ahead and start this one.

00:03 So the first thing we're supposed to do is draw a scatter diagram.

00:05 So the way i'm going to do that is i'm going to open a google sheet.

00:08 I'm going to type in my x and y values.

00:10 So it takes a few seconds to start off with, but i think it's definitely good for the long run.

00:15 So let's go and start that.

00:32 Okay, so there is the x's.

00:35 It might be a little bit more complex, but that's okay.

01:15 All right, so there are all our data points.

01:18 There, i think we got a moment.

01:20 Perfect.

01:20 Okay, cool.

01:21 So there are our data points.

01:22 So now it's time to start with part a and get our scatter diagram.

01:26 All right, and they do have some restrictions on the scatter diagram that we'll need to put in, but that's okay.

01:32 We'll go ahead and take care of that in just a second.

01:34 All righty, so i'm going to do insert chart here, and there's our scatter diagram.

01:41 It does say in the text that it wants my x to start at go from zero to 30.

01:48 So that's not that hard to do.

01:50 So we can just say customize here and then go for horizontal axis, start it at zero and end it at 30.

01:58 There we go.

01:59 And i can do the same thing for my vertical axis.

02:02 But i think the vertical axis is at zero to 20.

02:05 So that's an easy way to do that.

02:07 So again, how did i do that? fairly simple.

02:10 So what you're going to do is you're going to click on the chart there.

02:14 If you double click on the chart or hit the three buttons the upper right and hit edit chart, you're going to have your chart editor.

02:20 You're going to go to customize horizontal axis, and then you can change your min or your max, and then you can go to vertical axis, also change your min and your max.

02:31 And so there is our first scatter plot there.

02:36 The next thing they want us to do for part b is do our linear correlation coefficient.

02:41 Okay, and so by the way, guys, i'm going to change this chart in just a second.

02:47 So yeah, i'm going to delete this in just a second.

02:49 So make sure you get a copy down or take a screenshot or pause the video or something.

02:52 So our first linear correlation coefficient, this is going to be for part a.

03:01 First thing i'm going to do is the way you do a linear correlation coefficient is you start a new cell, hit the equal sign on your keyboard, type in the six letters c -o -r -r -e -l, open a parentheses.

03:12 Then you're going to highlight all your x values or explanatory variables, and then comma, and then highlight all your y values or response variables.

03:21 Close the parentheses, hit enter, and there is our linear correlation coefficient.

03:26 That's actually for part b.

03:28 I'm sorry.

03:28 That was a mistake.

03:29 For part b.

03:30 Okay, cool.

03:31 All right, so there is our data there.

03:34 Okay, so now what we're going to do is we're going to multiply both x and y by two.

03:40 Okay.

03:41 So, yeah, so that is my, let's see, i'm going to go ahead and make this smaller here.

03:48 There we go.

03:49 And i'm going to bring it down this way, at least over here, now.