CHAPTER 3

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CHAPTER 3

SOLUTIONS TO PROBLEMS

The 50 states of the U.S. are the units of analysis.
Months of the year are the units of analysis.
Marital status is a nominal level variable; the categories imply no order. For variables of this nature, bar chart and pie charts would be appropriate types of graphical presentation. Both would generally tell us that those individuals who had never been married had higher victimization rates compared to all other marital status categories.
The first step in constructing a grouped frequency distribution is to place all of the rates into ascending order. This is done simply by reordering your data from the lowest to the highest value. After the data is placed in ascending order, we must next calculate the range of our data and then decide on the number of intervals necessary to describe the distribution - the rule of thumb is usually between 10 to 20 intervals. The range of a set of scores is defined as the difference between the largest and the smallest values. In this distribution, motor vehicle theft rates range from 114 to 1038 is equal to a range of 924. Since the data range is almost 1,000, an appropriate number of intervals to select would be 10 ranging from 100 throug h 1100. After the intervals are created, you can now easily tally up the number of violent crime rates that fall within each of the intervals. This number is called frequency and is usually denoted with a small f. The sum of your interval fr equencies should equal your total sample size (n) which is 50. In addition to the frequency, you should also include a percentage column displayed in the table as shown below:
```
        Grouped Frequency Distribution for 
Motor vehicle theft rates for each state, UCR 1991.
_________________________________________________________
               Class 
Interval        Interval                f          Percent         
1               101-200                 8               16
2               201-300                 7               14
3               301-400                 9               18
4               401-500                 6               12      
5               501-600                 5               10
6               601-700                 5               10
7               701-800                 4                8
8               801-900                 1                2
9               901-1000                3                6
10              1001-1100               2                4
_________________________________________________________
                                         n=50           100%
```
A histogram constructed from the above grouped frequency distribution of motor vehicle theft rates would appear similar to the following:
The histogram of state-level motor vehicle theft rates reveals a distribution which is positively skewed. Most of the state rates cluster around the lower end of the distribution with a few states dragging the high end of the distribution out.

Both theft and violent rates of victimization for 1973 to 1990 are plotted below. The plots indicate that rates of theft victimization against 25-34 year olds increased during the early 1070s and then began a steady decline. Rates of violence during this time period have remained relatively stable.

             Crimes of               Crimes of 
              Violence                Theft
_____________________________________________
1973            34                      99
1974            38                      106
1975            39                      109
1976            40                      113
1977            42                      115
1978            40                      117
1979            44                      108
1980            40                      99
1981            44                      101
1982            46                      98
1983            41                      88
1984            38                      84
1985            37                      83
1986            34                      76
1987            34                      81
1988            35                      82
1989            35                      84
1990            36                      77
_____________________________________________

Note: Original data obtained from Criminal Victimization in the United States: 1973-90 Trends, Bureau of Justice Statistics has been rounded for purposes here.

The units of analysis in this table are years.

The original theft rates and smoothed rates are displayed below:

                       Original Rates          Smoothed  
                        of Theft                 Rates
_____________________________________________
1973                        99                     99
1974                       106                    104.7
1975                       109                    109.3
1976                       113                    112.3
1977                       115                    115
1978                       117                    113.3
1979                       108                    108
1980                        99                    102.7
1981                       101                     99.3
1982                        98                     95.7
1983                        88                     90
1984                        84                     85
1985                        83                     81
1986                        76                     80
1987                        81                     79.7
1988                        82                     82.3
1989                        84                     81
1990                        77                     77

_____________________________________________
Note: Original data obtained from Criminal Victimization in the United States: 1973-90 Trends, Bureau of Justice Statistics has been rounded for purposes here.

A time plot of both original rates and smoothed rates is presented below. Because there are no severe rate aberrations within the original data points, there is not much difference between the time plot for original and smoothed rates. The smoothed trend line, however, does reveal a somewhat smoother transition compared to the original data points.

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