Does prison really deter violent crime? Let x represent percent change in the rate of violent crime and y represent percent change in the rate of imprisonment in the general U.S. population. For 7 recent years, the following data have been obtained. x 6.1 5.6 4.0 5.2 6.2 6.5 11.1 y −1.8 −4.0 −7.2 −4.0 3.6 −0.1 −4.4 Complete parts (a) through (e), given Σx = 44.7, Σy = −17.9, Σx2 = 315.51, Σy2 = 119.41, Σxy = −110.15, and r ≈ 0.0883. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample correlation coefficient r. (Round your value for r to four decimal places.) Σx = Σy = Σx2 = Σy2 = Σxy = r = (c) Find x, and y. Then find the equation of the least-squares line = a + bx. (Round your answer to four decimal places.) x = y = = + x (e) Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r2 to four decimal places. Round your answers for the percentages to two decimal place.) r2 = explained % unexplained % (f) Considering the values of r and r2, does it make sense to use the least-squares line for prediction? Explain your answer. The correlation between the variables is so high that it does not make sense to use the least-squares line for prediction. The correlation between the variables is so low that it does not make sense to use the least-squares line for prediction. The correlation between the variables is so high that it makes sense to use the least-squares line for prediction. The correlation between the variables is so low that it makes sense to use the least-squares line for prediction.
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
Does prison really deter violent crime? Let x represent percent change in the rate of violent crime and y represent percent change in the rate of imprisonment in the general U.S. population. For 7 recent years, the following data have been obtained.
| x | 6.1 | 5.6 | 4.0 | 5.2 | 6.2 | 6.5 | 11.1 |
| y |
−1.8
|
−4.0
|
−7.2
|
−4.0
|
3.6 |
−0.1
|
−4.4
|
Complete parts (a) through (e), given Σx = 44.7,
Σx2 = 315.51, Σy2 = 119.41,
and r ≈ 0.0883.
(a) Draw a
(b) Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample
| Σx = | |
| Σy = | |
| Σx2 = | |
| Σy2 = | |
| Σxy = | |
| r = |
|
(c) Find x, and y. Then find the equation of the least-squares line = a + bx. (Round your answer to four decimal places.)
| x | = | |
| y | = | |
| = | + x |
(e) Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r2 to four decimal places. Round your answers for the percentages to two decimal place.)
| r2 = | |
| explained | % |
| unexplained | % |
(f) Considering the values of r and r2, does it make sense to use the least-squares line for prediction? Explain your answer.
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