The table shows the estimated number of lines of code written by computer programmers per hour when [tex]$x$[/tex] people are working.
Productivity
| People working | Lines of code written hourly |
| :------------- | :--------------------------- |
| 2 | 50 |
| 4 | 110 |
| 6 | 160 |
| 8 | 210 |
| 10 | 270 |
| 12 | 320 |
Which model best represents the data?
[tex]$y=47(1.191)^x$[/tex]
[tex]$y=34(1.204)^x$[/tex]
[tex]$y=26.9 x-1.3$[/tex]
[tex]$y=27 x-4$[/tex]
Answer (2)
The model that best represents the data is y = 26.9 x − 1.3 , as it has the smallest sum of squared errors compared to the other models evaluated. This model accurately captures the relationship between the number of people working and the lines of code written.
;
Calculate the sum of squared errors for each model using the given data points.
Compare the sum of squared errors for each model.
Identify the model with the smallest sum of squared errors.
Conclude that the model y = 26.9 x − 1.3 best represents the data, as it has the smallest sum of squared errors.
y = 26.9 x − 1.3
Explanation
Problem Analysis We are given a table of data showing the estimated number of lines of code written by computer programmers per hour when x people are working. We need to determine which of the four given models best represents the data. The data points are (2, 50), (4, 110), (6, 160), (8, 210), (10, 270), (12, 320). The four models are:
Model 1: y = 47 ( 1.191 ) x Model 2: y = 34 ( 1.204 ) x Model 3: y = 26.9 x − 1.3 Model 4: y = 27 x − 4
To determine which model best represents the data, we will calculate the sum of squared errors for each model. The model with the smallest sum of squared errors is the best fit.
Calculating Sum of Squared Errors For each model, we calculate the predicted y-values for each x-value in the table and then calculate the squared error for each data point: ( y p re d i c t e d − y a c t u a l ) 2 . Finally, we sum the squared errors to get the total sum of squared errors for each model.
Using a python calculation tool, the sum of squared errors for each model is as follows:
Model 1: Sum of Squared Errors = 5524.7386102429155 Model 2: Sum of Squared Errors = 11016.094163149735 Model 3: Sum of Squared Errors = 42.69999999999972 Model 4: Sum of Squared Errors = 60
Comparing Models Comparing the sum of squared errors for each model, we see that Model 3 has a sum of squared errors of approximately 42.7, and Model 4 has a sum of squared errors of 60. Since 42.7 is less than 60, Model 3 has the smallest sum of squared errors. Therefore, Model 3 best represents the data.
Final Answer The model that best represents the data is y = 26.9 x − 1.3 .
Examples
In software development, understanding the relationship between the number of developers and the lines of code produced can help project managers estimate project timelines and allocate resources effectively. For instance, if a project requires writing 10,000 lines of code, the model can help estimate how many developers are needed and how long it will take to complete the project. This type of analysis ensures projects are completed on time and within budget, optimizing team productivity and resource allocation.
