(a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and (c

(a) draw a scatter diagram of the data, (b) by hand, compute...

Key Concepts

Scatter Diagram

A scatter diagram is a type of graph used to visually represent the relationship between two quantitative variables by plotting data points on a two-dimensional grid. Each point on the plot corresponds to an x-y pair from the data set, allowing one to observe patterns, trends, and potential outliers. It is a fundamental tool in exploratory data analysis to quickly assess the nature of the relationship between variables.

Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It is computed using the covariance of the variables divided by the product of their standard deviations, and its value ranges between -1 and 1. A value close to 1 indicates a strong positive linear relationship, a value near -1 indicates a strong negative linear relationship, and a value around 0 suggests little to no linear relationship.

Relationship Analysis

Relationship analysis involves examining the pattern of the data points in the scatter diagram and calculating the correlation coefficient to determine the form of the relationship between the variables. A positive correlation indicates that as one variable increases, the other tends to increase as well, while a negative correlation would imply the opposite. Visual analysis and the computed correlation help to ascertain whether the relationship is linear and to identify any anomalies in the data.

Transcript

00:01 All righty guys.

00:02 So for problem 22, we need to start by drawing a scatter diagram of the data.

00:07 I went ahead and took the liberty of using example 2 from our textbook, kind of the template to solve the problem.

00:13 And so, yeah, you guys can look at example 2 in the text if you want.

00:16 But in order to get the scatter plot done for part a, i need to go ahead and input the data.

00:20 So the x values are 0, 1, 1, 2, 4, and 7.

00:26 And my y values are going to be three, five, four, six, eight, and nine.

00:33 Okay.

00:34 So to get my scatterplot, the one i'm going to do is going to highlight all of that.

00:38 I'm going to do, let's see, insert chart, and there we go.

00:44 There is my chart for that.

00:47 Okay, perfect.

00:48 Awesome.

00:48 Okay, so with that, there's my chart.

00:52 So i'm going to give you guys a second or two to copy that down.

00:55 I'm going to give you a second or two to copy that down or take a screenshot because i'm going to be inputting more data on my i'm going to be putting more data on my spreadsheet.

01:06 And if i put more data in, it might mess up the chart.

01:08 So go ahead and take a screenshot or copy that down really quick before i delete it because i'm going to delete it in a second.

01:15 Before i do delete it, i am going to take a quick comment here.

01:18 Sometimes what i like to do is when i do the scatter plot, i like to use that scatter plot to take a guess to, to, to, take a guess about what the correlation coefficient is going to be.

01:30 So looking at the data here, it does seem like it follows kind of a linear pattern.

01:34 I think i can pretty easily draw a line right down the middle of that guy.

01:38 And all the data points would be pretty close to that line.

01:41 And so i would actually assume, and since that line's going up, by the way, i'm going to assume there's a pretty strong positive correlation.

01:47 So somewhere around, i don't know, 0 .9 or something like that, it's kind of hard to say exactly what it will be.

01:53 But i would say it's pretty close to one.

01:55 Maybe in the 0 .9s or it might be 0 .8s.

01:58 We'll see.

01:58 But anyhow, so hopefully that gave you guys enough time to copy that down or take a screenshot.

02:02 But i'm going to delete this.

02:03 And i'll move on to part b.

02:06 Okay.

02:06 So the next thing i need to do is this.

02:09 First, to find the linear correlation coefficient, tongue twister, i need to find the means here.

02:17 And so my mean, for my intents and purposes, i'm going to call the mean of the x values capital x.

02:23 Means the y values capital y.

02:25 How do i do that in a google sheet? very simple.

02:27 Start by opening a new cell, hitting the equal symbol on your keyboard, and then type in the word average, a -v -e -r -a -g -e.

02:39 Open parentheses, highlight all the text that you want, and close parentheses, hit enter, and there you go.

02:48 And then we can do the same for y.

02:50 Sometimes, excuse me, sometimes the google sheet will guess it for you of what they want.

02:56 And then, yeah, and if it guesses it for you, great.

02:59 More power to you.

03:00 Click on it, hit enter, and you're done.

03:01 The next thing we need to do is to find the standard deviation.

03:04 How you find the standard deviation, very similar to finding the mean or the average.

03:08 You write equals, and then the letters, stb -b -e -v, open parentheses, select all the values you want.

03:16 Don't do the mean because the mean's not actually a data point.

03:18 Close parentheses, and there we go.

03:20 There's standard deviation.

03:21 Sometimes it will.

03:22 Guess what you want.

03:23 In this case, it does guess it.

03:25 Great.

03:25 I click on that and there we go.

03:28 Easy peasy.

03:29 The next thing we need to do to find a linear correlation coefficient is this.

03:35 Okay, so the first thing we need to do is we need to find that, so we already found the mean of the standard deviation.

03:42 Now we need to find the z scores.

03:43 How do you find z scores? take each data point, subtract by the mean, divide by the standard deviation.

03:48 All right, so let's do the x's really quickly.

03:50 So what we'll do is this.

03:51 We'll take the first data point, subtract the mean, and divide by the standard deviation.

03:59 Very easy.

04:00 Let's keep on going.

04:01 First take a beta point, subtract the mean, and divide by the standard deviation...

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