standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: Task
Many counties in the United States are governed by a county council. At public county council meetings, county residents are usually allowed to bring up issues of concern. At a recent public County Council meeting, one resident expressed concern that 3 new coffee shops from a popular coffee shop chain were planning to open in the county, and the resident believed that this would create an increase in property crimes in the county. (Property crimes include burglary, larceny-theft, motor vehicle theft, and arson -- From http://www.fbi.gov/about-us/cjis/ucr/crime-in-the-u.s/2010/crime-in-the-u.s.-2010/property-crime accessed on December 5, 2012.)
To support this claim, the resident presented the following data and scatterplot (with the least-squares line shown) for 8 counties in the state:
{"table": [{"_attributes": {"class": ["centered", "narrow"]}, "tbody": [{"tr": [{"th": [{"_value": "County"}, {"_value": "Shops"}, {"_value": "Crimes"}]}, {"td": [{"_value": "A"}, {"_value": "9"}, {"_value": "4000"}]}, {"td": [{"_value": "B"}, {"_value": "1"}, {"_value": "2700"}]}, {"td": [{"_value": "C"}, {"_value": "0"}, {"_value": "500"}]}, {"td": [{"_value": "D"}, {"_value": "6"}, {"_value": "4200"}]}, {"td": [{"_value": "E"}, {"_value": "15"}, {"_value": "6800"}]}, {"td": [{"_value": "F"}, {"_value": "50"}, {"_value": "20800"}]}, {"td": [{"_value": "G"}, {"_value": "5"}, {"_value": "2800"}]}, {"td": [{"_value": "H"}, {"_value": "24"}, {"_value": "15400"}]}]}]}]}
[IMG]
The scatterplot shows a positive linear relationship between "Shops" (the number of coffee shops of this coffee shop chain in the county) and "Crimes" (the number of annual property crimes for the county). In other words, counties with more of these coffee shops tend to have more property crimes annually.
Does the relationship between Shops and Crimes appear to be linear? Would you consider the relationship between Shops and Crimes to be strong, moderate, or weak?
Compute the correlation coefficient.  Does the value of the correlation coefficient support your choice in part (a)? Explain.
The equation of the least-squares line for these data is:
$$ \text{Predicted Crimes} = 1434 + 415.7 \text{(Shops)} $$
Based on this line, what is the estimated number of additional annual property crimes for a given county that has 3 more coffee shops than another county?
Do these data support the claim that building 3 additional coffee shops will necessarily cause an increase in property crimes?  What other variables might explain the positive relationship between the number of coffee shops for this coffee shop chain and the number of annual property crimes for these counties?
If the following two counties were added to the data set, would you still consider using a line to model the relationship? If not, what other types (forms) of model would you consider?
{"table": [{"_attributes": {"class": ["centered", "narrow"]}, "tbody": [{"tr": [{"th": [{"_value": "County"}, {"_value": "Shops"}, {"_value": "Crimes"}]}, {"td": [{"_value": "I"}, {"_value": "25"}, {"_value": "36900"}]}, {"td": [{"_value": "J"}, {"_value": "27"}, {"_value": "24100"}]}]}]}]}


aligns: no