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?
A. [tex]$y=47(1.191)^x$[/tex]
B. [tex]$y=34(1.204)^x$[/tex]
C. [tex]$y=26.9 x-1.3$[/tex]
D. [tex]$y=27 x-4$[/tex]
Answer (2)
Evaluate each model by calculating the predicted y-values for each given x-value.
Calculate the sum of squared differences between the predicted and actual y-values for each model.
Compare the sums of squared differences to determine the best fit.
The model with the smallest sum of squared differences is y = 26.9 x − 1.3 , so the final answer is y = 26.9 x − 1.3 .
Explanation
Understanding the Problem We are given a table of data showing the relationship between the number of people working on a software project and the number of lines of code written per hour. We are also given four mathematical models and asked to determine which model best represents the data.
Solution Strategy To determine which model best represents the data, we will calculate the predicted y-values for each model at each given x-value. Then, we will calculate the sum of squared differences between the predicted y-values and the actual y-values for each model. The model with the smallest sum of squared differences will be considered the best fit.
Evaluating Model 1: y = 47 ( 1.191 ) x Let's evaluate Model 1: y = 47 ( 1.191 ) x for each x-value in the table. For x=2: y = 47 ( 1.191 ) 2 = 47 ( 1.418481 ) ≈ 66.67 For x=4: y = 47 ( 1.191 ) 4 = 47 ( 2.01208 ) ≈ 94.57 For x=6: y = 47 ( 1.191 ) 6 = 47 ( 2.8538 ) ≈ 134.13 For x=8: y = 47 ( 1.191 ) 8 = 47 ( 4.0475 ) ≈ 190.23 For x=10: y = 47 ( 1.191 ) 10 = 47 ( 5.741 ) ≈ 269.83 For x=12: y = 47 ( 1.191 ) 12 = 47 ( 8.140 ) ≈ 382.58 Now, let's calculate the sum of squared differences for Model 1: ( 66.67 − 50 ) 2 + ( 94.57 − 110 ) 2 + ( 134.13 − 160 ) 2 + ( 190.23 − 210 ) 2 + ( 269.83 − 270 ) 2 + ( 382.58 − 320 ) 2 ≈ 277.89 + 238.15 + 663.88 + 390.85 + 0.03 + 3916.76 ≈ 5487.56
Evaluating Model 2: y = 34 ( 1.204 ) x Now let's evaluate Model 2: y = 34 ( 1.204 ) x for each x-value in the table. For x=2: y = 34 ( 1.204 ) 2 = 34 ( 1.449616 ) ≈ 49.29 For x=4: y = 34 ( 1.204 ) 4 = 34 ( 2.1013 ) ≈ 71.44 For x=6: y = 34 ( 1.204 ) 6 = 34 ( 3.045 ) ≈ 103.53 For x=8: y = 34 ( 1.204 ) 8 = 34 ( 4.415 ) ≈ 150.11 For x=10: y = 34 ( 1.204 ) 10 = 34 ( 6.39 ) ≈ 217.26 For x=12: y = 34 ( 1.204 ) 12 = 34 ( 9.25 ) ≈ 314.5 Now, let's calculate the sum of squared differences for Model 2: ( 49.29 − 50 ) 2 + ( 71.44 − 110 ) 2 + ( 103.53 − 160 ) 2 + ( 150.11 − 210 ) 2 + ( 217.26 − 270 ) 2 + ( 314.5 − 320 ) 2 ≈ 0.50 + 1486.11 + 3181.74 + 3586.69 + 2781.4 + 30.25 ≈ 11066.59
Evaluating Model 3: y = 26.9 x − 1.3 Now let's evaluate Model 3: y = 26.9 x − 1.3 for each x-value in the table. For x=2: y = 26.9 ( 2 ) − 1.3 = 53.8 − 1.3 = 52.5 For x=4: y = 26.9 ( 4 ) − 1.3 = 107.6 − 1.3 = 106.3 For x=6: y = 26.9 ( 6 ) − 1.3 = 161.4 − 1.3 = 160.1 For x=8: y = 26.9 ( 8 ) − 1.3 = 215.2 − 1.3 = 213.9 For x=10: y = 26.9 ( 10 ) − 1.3 = 269 − 1.3 = 267.7 For x=12: y = 26.9 ( 12 ) − 1.3 = 322.8 − 1.3 = 321.5 Now, let's calculate the sum of squared differences for Model 3: ( 52.5 − 50 ) 2 + ( 106.3 − 110 ) 2 + ( 160.1 − 160 ) 2 + ( 213.9 − 210 ) 2 + ( 267.7 − 270 ) 2 + ( 321.5 − 320 ) 2 ≈ 6.25 + 13.69 + 0.01 + 15.21 + 5.29 + 2.25 ≈ 42.7
Evaluating Model 4: y = 27 x − 4 Now let's evaluate Model 4: y = 27 x − 4 for each x-value in the table. For x=2: y = 27 ( 2 ) − 4 = 54 − 4 = 50 For x=4: y = 27 ( 4 ) − 4 = 108 − 4 = 104 For x=6: y = 27 ( 6 ) − 4 = 162 − 4 = 158 For x=8: y = 27 ( 8 ) − 4 = 216 − 4 = 212 For x=10: y = 27 ( 10 ) − 4 = 270 − 4 = 266 For x=12: y = 27 ( 12 ) − 4 = 324 − 4 = 320 Now, let's calculate the sum of squared differences for Model 4: ( 50 − 50 ) 2 + ( 104 − 110 ) 2 + ( 158 − 160 ) 2 + ( 212 − 210 ) 2 + ( 266 − 270 ) 2 + ( 320 − 320 ) 2 ≈ 0 + 36 + 4 + 4 + 16 + 0 ≈ 60
Comparing the Models Comparing the sums of squared differences, we have: Model 1: 5487.56 Model 2: 11066.59 Model 3: 42.7 Model 4: 60 The smallest sum of squared differences is for Model 3.
Final Answer Therefore, the model that best represents the data is y = 26.9 x − 1.3 .
Examples
In software development, understanding the relationship between team size and productivity is crucial for project planning. This problem demonstrates how to use mathematical models to represent and analyze this relationship. For instance, a project manager can use these models to estimate the number of lines of code a team can produce in a given time frame, helping them allocate resources effectively and set realistic deadlines. By fitting a model to historical data, they can predict future productivity based on team size, allowing for better decision-making and resource allocation. This approach ensures projects are well-planned and executed efficiently, maximizing the team's output and minimizing potential delays.
The model that best represents the data from the table regarding the number of lines of code written per hour by computer programmers is y = 26.9 x − 1.3 . This model yields the smallest sum of squared differences when compared to the actual data values. Thus, it is the most accurate representation of the relationship between the number of people working and their productivity in lines of code written hourly.
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