MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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### Problem Statement **(b)** Verify the given sums \(\sum x\), \(\sum y\), \(\sum x^2\), \(\sum y^2\), \(\sum xy\), and the value of the sample correlation coefficient \(r\). (For each answer, enter a number. Round your value for \(r\) to three decimal places.) - \(\sum x =\) \_\_\_\_\_ - \(\sum y =\) \_\_\_\_\_ - \(\sum x^2 =\) \_\_\_\_\_ - \(\sum y^2 =\) \_\_\_\_\_ - \(\sum xy =\) \_\_\_\_\_ - \(r =\) \_\_\_\_\_ **(c)** Find \(\bar{x}\) and \(\bar{y}\). Then find the equation of the least-squares line \(\hat{y} = a + bx\). (For each answer, enter a number. Round your answers for \(\bar{x}\) and \(\bar{y}\) to two decimal places. Round your answers for \(a\) and \(b\) to three decimal places.) - \(\bar{x} =\) \_\_\_\_\_ - \(\bar{y} =\) \_\_\_\_\_ - \(\hat{y} =\) \_\_\_\_\_ + \_\_\_\_\_ \(x\) **(d)** Graph the least-squares line. Be sure to plot the point \((\bar{x}, \bar{y})\) as a point on the line. (Select the correct graph.) **Graphs Explanation:** Two choices labeled "Choice A" and "Choice B" are provided: - **Choice A:** A graph with axes labeled \(y\) ranging from 0 to 200. It potentially shows a line which seems to be a candidate for the least-squares line. - **Choice B:** Another graph similarly labeled with axes, showing a different potential line for the least-squares method. The correct graph should include the plotted point \((\bar{x}, \bar{y})\) on the line.
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Transcribed Image Text:### Problem Statement **(b)** Verify the given sums \(\sum x\), \(\sum y\), \(\sum x^2\), \(\sum y^2\), \(\sum xy\), and the value of the sample correlation coefficient \(r\). (For each answer, enter a number. Round your value for \(r\) to three decimal places.) - \(\sum x =\) \_\_\_\_\_ - \(\sum y =\) \_\_\_\_\_ - \(\sum x^2 =\) \_\_\_\_\_ - \(\sum y^2 =\) \_\_\_\_\_ - \(\sum xy =\) \_\_\_\_\_ - \(r =\) \_\_\_\_\_ **(c)** Find \(\bar{x}\) and \(\bar{y}\). Then find the equation of the least-squares line \(\hat{y} = a + bx\). (For each answer, enter a number. Round your answers for \(\bar{x}\) and \(\bar{y}\) to two decimal places. Round your answers for \(a\) and \(b\) to three decimal places.) - \(\bar{x} =\) \_\_\_\_\_ - \(\bar{y} =\) \_\_\_\_\_ - \(\hat{y} =\) \_\_\_\_\_ + \_\_\_\_\_ \(x\) **(d)** Graph the least-squares line. Be sure to plot the point \((\bar{x}, \bar{y})\) as a point on the line. (Select the correct graph.) **Graphs Explanation:** Two choices labeled "Choice A" and "Choice B" are provided: - **Choice A:** A graph with axes labeled \(y\) ranging from 0 to 200. It potentially shows a line which seems to be a candidate for the least-squares line. - **Choice B:** Another graph similarly labeled with axes, showing a different potential line for the least-squares method. The correct graph should include the plotted point \((\bar{x}, \bar{y})\) on the line.
### Understanding Calf Weight and Age at Bar-S Cattle Ranch You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are considering purchasing some to add to the Bar-S herd. A fundamental aspect you are investigating is the relationship between age and weight in these calves. Specifically, how much should a healthy calf weigh at different ages? #### Data Overview The age (\(x\)) and weight (\(y\)) data for the calves is as follows: | Age (weeks) \(x\) | Weight (kg) \(y\) | |-------------------|-------------------| | 1 | 39 | | 5 | 47 | | 11 | 73 | | 16 | 100 | | 26 | 150 | | 36 | 200 | #### Statistical Summary - Sum of ages: \(\sum x = 95\) - Sum of weights: \(\sum y = 609\) - Sum of ages squared: \(\sum x^2 = 2375\) - Sum of weights squared: \(\sum y^2 = 81,559\) - Sum of products of ages and weights: \(\sum x y = 13,777\) - Correlation coefficient: \(r \approx 0.997\) #### Graphical Representation To visualize the relationship between age and weight, examine the scatter plot diagrams. Choose the correct graph from the options shown: - **Choice A:** Displays a scatter plot with a cluttered data pattern. - **Choice B:** Portrays a clear linear relationship with increasing age associated with increasing weight. - **Choice C and D:** Demonstrate scattered data with less apparent trends. The scatter plot in **Choice B** accurately represents the data: as the age of calves increases, their weight also increases in a roughly linear pattern. This positive correlation is confirmed by a correlation coefficient (\(r\)) very close to 1, indicating a strong linear relationship. #### Conclusion Understanding these data patterns helps in determining growth expectations for the calves, guiding the decision on which ones to purchase for optimal growth of the herd at Bar-S Ranch.
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Transcribed Image Text:### Understanding Calf Weight and Age at Bar-S Cattle Ranch You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are considering purchasing some to add to the Bar-S herd. A fundamental aspect you are investigating is the relationship between age and weight in these calves. Specifically, how much should a healthy calf weigh at different ages? #### Data Overview The age (\(x\)) and weight (\(y\)) data for the calves is as follows: | Age (weeks) \(x\) | Weight (kg) \(y\) | |-------------------|-------------------| | 1 | 39 | | 5 | 47 | | 11 | 73 | | 16 | 100 | | 26 | 150 | | 36 | 200 | #### Statistical Summary - Sum of ages: \(\sum x = 95\) - Sum of weights: \(\sum y = 609\) - Sum of ages squared: \(\sum x^2 = 2375\) - Sum of weights squared: \(\sum y^2 = 81,559\) - Sum of products of ages and weights: \(\sum x y = 13,777\) - Correlation coefficient: \(r \approx 0.997\) #### Graphical Representation To visualize the relationship between age and weight, examine the scatter plot diagrams. Choose the correct graph from the options shown: - **Choice A:** Displays a scatter plot with a cluttered data pattern. - **Choice B:** Portrays a clear linear relationship with increasing age associated with increasing weight. - **Choice C and D:** Demonstrate scattered data with less apparent trends. The scatter plot in **Choice B** accurately represents the data: as the age of calves increases, their weight also increases in a roughly linear pattern. This positive correlation is confirmed by a correlation coefficient (\(r\)) very close to 1, indicating a strong linear relationship. #### Conclusion Understanding these data patterns helps in determining growth expectations for the calves, guiding the decision on which ones to purchase for optimal growth of the herd at Bar-S Ranch.
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