10. Table 4.31 gives the relationship between the total number of firearms (TFA) and the number of aggravated assaults with firearms (AWF) in 4 US states and the District of Columbia in 2010.
Table 4.31
DC TN DE SC GA
TFA 2268 13132 1701 8137 11728
AWF 606 8231 824 5274 5160
The regression line is AWF = -361.102 + 0.5925 TFA, and the linear correlation coefficient is r = 0.9575.
a) What are the predictor and response variables?
predictor: AWF
response: TFA
b) Estimate the number of aggravated assaults with firearms if the total number of firearms is 5000. If the estimate is not possible, explain why not.
2,601.398
c) Estimate the number of aggravated assaults with firearms if the total number of firearms is 400. If the estimate is not possible, explain why not.
-124.102, it is not possible because it is a negative number
d) Estimate the number of firearms if the number of aggravated assaults is 720. If the estimate is not possible, explain why not.
65.498
Can you explain how you got these answers?
To estimate the number of aggravated assaults with firearms (AWF) given a certain number of total firearms (TFA), we can use the regression line provided: AWF = -361.102 + 0.5925 TFA.
a) The predictor variable is AWF, representing the number of aggravated assaults with firearms. The response variable is TFA, representing the total number of firearms.
b) To estimate the number of aggravated assaults with firearms if the total number of firearms is 5000, we substitute 5000 for TFA in the regression line equation:
AWF = -361.102 + 0.5925 * 5000
AWF = -361.102 + 2962.5
AWF = 2601.398
Therefore, the estimated number of aggravated assaults with firearms is 2601.398 when the total number of firearms is 5000.
c) To estimate the number of aggravated assaults with firearms if the total number of firearms is 400, we again substitute 400 for TFA in the regression line equation:
AWF = -361.102 + 0.5925 * 400
AWF = -361.102 + 237
AWF = -124.102
However, we cannot have a negative value for the number of aggravated assaults, so it is not possible to estimate the number of aggravated assaults with firearms if the total number of firearms is 400.
d) To estimate the number of firearms if the number of aggravated assaults is 720, we rearrange the regression line equation to solve for TFA:
AWF = -361.102 + 0.5925 TFA
0.5925 TFA = AWF + 361.102
TFA = (AWF + 361.102) / 0.5925
Substituting AWF = 720 into the equation:
TFA = (720 + 361.102) / 0.5925
TFA = 1081.102 / 0.5925
TFA = 1826.498
Therefore, the estimated number of firearms is 1826.498 when the number of aggravated assaults is 720.