Branches of mechanical engineering: Frequency As Well As Chi-Square Exam For Independence Exercises + Solutions - R

(This article was starting fourth dimension published on R-exercises, together with kindly contributed to R-bloggers)

In this exercise, nosotros encompass around basics on frequency tables. We besides briefly expect at chi-square exam for independence to give away relationships of 2 variables. Before proceeding, it mightiness live on helpful to expect over the attention pages for the tablesummary, and margin.table functions.
Answers to the exercises are available here.
If yous obtained a dissimilar (correct) respond than those listed on the solutions page, delight experience gratuitous to post service your respond equally a comment on that page.
Exercise 1
utilisation the attach() command to charge the trees dataset inward R
Exercise 2
Use the table() command alongside the arguments: trees$Height together with trees$Volume. This volition generate a two-way frequency table. Store this inward variable mytable.
Exercise 3
If yous are familiar alongside excel pin tables, hence yous volition know this function. Use the margin.table( ) role to snuff it the Height frequencies summed over Volume
Exercise 4
Use the margin.table( ) function to snuff it the Volume frequencies summed over Height.
Exercise 5
Now utilisation the table() function 1 time again precisely using all the features of the trees dataset, that includes girth, summit together with volume. This volition impress out a multidimensional iii means frequency table.
Exercise 6
Suppose yous convey a variable ‘a’ that stores a 2d sample of heights of trees.

a=c(70, 65, 63, 72, 80, 83, 66, 75, 80, 75, 79, 76, 76, 69, 75, 74, 85, 8, 71, 63, 78, 80, 74, 72, 77, 81, 82, 80, 86, 80, 87)

Use the cbind() to add together the a column to your trees dataset. Store the results dorsum into trees.
Exercise 7
Now do a 2 means frequency tabular array betwixt Height together with a equally the arguments. Store this tabular array inward mytable_2.
Exercise 8
Use the margin.table() role 1 time again from Q3 together with snuff it Height frequencies summertime over a. What differences do yous give away from the results of Q3.
Exercise 9
Chi Square exam for independance:
a)Print the results of the summary() role on mytable. Note the Chi Square exam for independance results together with P value
b)Print the results of the summary() role on mytable_2. Note the Chi Square exam for independance results together with P value.
Exercise 10
What did the chi foursquare exam for independance attention yous to see?


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Below are the solutions to these exercises on frequency together with chi-square test
############### #             # # Exercise 1  # #             # ############### attach(trees)  ############### #             # # Exercise 2  # #             # ###############  mytable=table(trees$Height,trees$Volume)  ############### #             # # Exercise iii  # #             # ###############  margin.table(mytable, 1) 
##  ## 63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87  ##  1  1  1  1  1  1  1  2  2  iii  2  1  1  1  v  2  1  1  1  1  1 
############### #             # # Exercise 4  # #             # ###############  margin.table(mytable, 2) 
##  ## 10.2 10.3 15.6 16.4 18.2 18.8 19.1 19.7 19.9   21 21.3 21.4 22.2 22.6 24.2  ##    1    2    1    1    1    1    1    1    1    1    1    1    1    1    1  ## 24.9 25.7 27.4 31.7 33.8 34.5 36.3 38.3 42.6   51 51.5 55.4 55.7 58.3   77  ##    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
############### #             # # Exercise v  # #             # ###############  table(trees$Girth,trees$Height, trees$Volume) 
## , ,  = 10.2 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 10.3 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 15.6 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 16.4 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 18.2 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 18.8 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 19.1 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 19.7 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 19.9 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 21 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 21.3 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 21.4 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 22.2 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 22.6 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 24.2 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 24.9 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 25.7 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.7  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xiv    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   14.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   sixteen    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   16.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   17.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   xviii    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   20.6  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ## , ,  = 27.4 ##  ##        ##        63 64 65 66 69 seventy 71 72 74 75 76 77 78 79 fourscore 81 82 83 85 86 87 ##   8.3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.6   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   8.8   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   10.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   eleven    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   11.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   12.9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0 ##   13.7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##   13.8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ##  ##  [ reached getOption("max.print") -- omitted 10 row(s) together with 12 matrix slice(s) ] 
###############  #             #  # Exercise half dozen  #  #             #  ###############
a=c(70, 65, 63, 72, 80, 83, 66, 75, 80, 75, 79, 76, 76, 69, 75, 74, 85, 8, 71, 63, 78, 80, 74, 72, 77, 81, 82, 80, 86, 80, 87) trees=cbind(trees,a)
############### # # # Exercise vii # # # ############### mytable_2=table(trees$Height,trees$a)
############### # # # Exercise 8 # # # ###############
##for mytable_2, nosotros are seeing values greater than 1 together with at that spot are to a greater extent than matching values than

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