Problem

Given are five observations for two variables, $x…

06:46

Problem 1 Easy Difficulty

Given are five observations for two variables, $x$ and $y$
$$
\frac{x_{i}|1 \quad 2 \quad 3 \quad 4 \quad 5}{y_{i}|3 \quad 7 \quad 5 \quad 11}
$$

a. Develop a scatter diagram for these data
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Try to approximate the relationship between $x$ and $y$ by drawing a straight line
through the data.
d. Develop the estimated regression equation by computing the values of $b_{0}$ and $b_{1}$ using equations $(12.6)$ and $(12.7) .$
e. Use the estimated regression equation to predict the value of $y$ when $x=4$ .

Answer

a) See the diagram, b) Positive linear relationship, c) See the diagram, d)
$\hat{y}=0.2+2.6 x,$ e) $\hat{y}=10.6$

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Chapter 12

Simple Linear Regression

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Video Transcript

This is a linear regression exercise and we are given the data shown in this Excel spreadsheet and for part they were asked to make a scatter diagram for the data to do this in Excel, we can simply select the two columns, click on insert select charts and a scatter diagram and this is our scattered diagram. If you're going to hand this in, you'd want to put an appropriate title and label the X and Y axes, so that's part A. And then for part B. We are asked what does the scatter plot indicate about the relationship between the two variables? And it looks like there's a linear pattern to these scatters and it has a positive slope. So we can say that the chart indicates that two variables are positively the nearly correlated. Now, for part C, we are asked to try to draw a line through the scatter to approximately to approximate the relationship between X and Y. So we can draw something. This isn't very straight line I'm drawing, but it's approximately like that. And if we want to estimate the slope can take the data between two points. And we can say that the rise over the run is so this would be the rise from this line of this line, so that's about 10. And the run would be from here to here, and that's about 3.5. So it seems like the slope, it is about three and the y intercept would be just above zero, maybe 0.5. And then for part B, we are asked to estimate the regression equation by using equations 12.6 and 12.7 in the textbook. And these are these equations 12.6 is the first one which gives you the slope, and 12.7 is the second one which gives you the intercept. So the first thing we should do is find the mean for X. And the mean for Y. Are the sample average for X. And the sample average for Y. We can do this in Excel by using the average function. Yeah. So that gives us three and eight. And now for the slope let's calculate the numerator and the denominator separately on the Excel spreadsheet. So in the numerator we have X I minus the average of the excess which is three times why I minus the average of the wise which is eight. And we want to do that for each I and then we want to sum them so that 26 is the numerator. And then for the denominator, just over here it's X I minus X I minus X. Bar which is three and that's squared. And then we do this for each X. I. And then some them and we get 10. So we have 26/10 which gives us 2.6. And then for the intercept it's why bar just eight minus the slope which is 2.6 times X. Bar which is three. And this comes out to 0.2. And so the equation for our aggression line is given as zero point two plus 2.6 times X. And lastly, we are asked to estimate why when X. Is equity for. So we simply plug X equals four into our regression equation and we get 10.6.

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