Bhagwan Mahavir College of
Management
BMEF Campus, VIP Road, Bharthana
Vesu, Surat.
Lab Manual
On
4649303– Statistical Methods (SM)
M.C.A. – 4th Semester
Submitted By : Submitted To : 58 – Paresh Chavda Prof. Harshad Patel
Relative frequency:
Q.1
data=c("android","java","sm","jwt","python","c","c++","php","Core
java")
> data
[1] "android" "java" "sm" "jwt" "python" "c" "c++"
[8] "php" "Core java"
data.freq=table(data)
> data.freq
data
android c c++ Core java java
jwt php python sm
1 1 1 1 1 1 1 1 1
> data.rel=data.freq/nrow(data.freq)
> data.rel
data
android c c++ Core java java
jwt php python sm
0.1111111 0.1111111 0.1111111 0.1111111
0.1111111 0.1111111 0.1111111 0.1111111 0.1111111
> data.per=data.rel*100
> data.per
data
android c c++ Core java java
jwt php python sm
11.11111
11.11111 11.11111 11.11111
11.11111 11.11111 11.11111
11.11111 11.11111
> barplot(data.freq)
> pie(data.freq)
> cbind(data.freq,data.rel,data.per)
data.freq data.rel data.per
android 1 0.1111111 11.11111
c 1 0.1111111 11.11111
c++ 1 0.1111111 11.11111
Core java 1 0.1111111 11.11111
java 1 0.1111111 11.11111
jwt 1 0.1111111 11.11111
php 1 0.1111111 11.11111
python 1 0.1111111 11.11111
sm 1 0.1111111 11.11111
Q.2
library(MASS)
>
data()
>
phones
$year
[1] 50 51 52 53 54 55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72 73
$calls
[1]
4.4 4.7 4.7
5.9 6.6 7.3
8.1 8.8 10.6
12.0 13.5 14.9
[13] 16.1
21.2 119.0 124.0 142.0 159.0 182.0 212.0
43.0 24.0 27.0
29.0
duration=phones$calls
>
duration
[1]
4.4 4.7 4.7
5.9 6.6 7.3
8.1 8.8 10.6
12.0 13.5 14.9
[13] 16.1
21.2 119.0 124.0 142.0 159.0 182.0 212.0
43.0 24.0 27.0
29.0
>
duration.freq=table(duration)
>
cbind(duration.freq)
duration.freq
4.4 1
4.7 2
5.9 1
6.6 1
7.3 1
8.1 1
8.8 1
10.6 1
12 1
13.5 1
14.9 1
16.1 1
21.2 1
24 1
27 1
29 1
43 1
119 1
124 1
142 1
159 1
182 1
212 1
>
duration.rel=duration.freq/nrow(duration.freq)
>
duration.rel
duration
4.4 4.7 5.9 6.6 7.3 8.1 8.8
0.04347826
0.08695652 0.04347826 0.04347826 0.04347826 0.04347826 0.04347826
10.6 12 13.5 14.9 16.1 21.2 24
0.04347826
0.04347826 0.04347826 0.04347826 0.04347826 0.04347826 0.04347826
27 29 43 119 124 142 159
0.04347826
0.04347826 0.04347826 0.04347826 0.04347826 0.04347826 0.04347826
182 212
0.04347826
0.04347826
> barplot(duration.freq)
>
pie(duration.freq)
>
cbind(duration.freq,duration.rel,duration.per)
duration.freq duration.rel duration.per
4.4 1 0.04347826 4.347826
4.7 2 0.08695652 8.695652
5.9 1 0.04347826 4.347826
6.6 1 0.04347826 4.347826
7.3 1 0.04347826 4.347826
8.1 1 0.04347826 4.347826
8.8 1 0.04347826 4.347826
10.6 1
0.04347826 4.347826
12 1 0.04347826 4.347826
13.5 1
0.04347826 4.347826
14.9 1
0.04347826 4.347826
16.1 1
0.04347826 4.347826
21.2 1
0.04347826 4.347826
24 1 0.04347826 4.347826
27 1 0.04347826 4.347826
29 1 0.04347826 4.347826
43 1 0.04347826 4.347826
119 1 0.04347826 4.347826
124 1 0.04347826 4.347826
142 1 0.04347826 4.347826
159 1 0.04347826 4.347826
182 1 0.04347826 4.347826
212 1 0.04347826 4.347826
Q-3
> library(MASS)
> data()
> shoes
$A
[1] 13.2 8.2 10.9 14.3 10.7 6.6
9.5 10.8 8.8 13.3
$B
[1] 14.0 8.8 11.2 14.2 11.8 6.4
9.8 11.3 9.3 13.6
> price=shoes$A
> price
[1] 13.2 8.2 10.9 14.3 10.7 6.6
9.5 10.8 8.8 13.3
> price.freq=table(price)
> price
[1] 13.2 8.2 10.9 14.3 10.7 6.6
9.5 10.8 8.8 13.3
> cbind(price.freq)
price.freq
6.6 1
8.2 1
8.8 1
9.5 1
10.7 1
10.8 1
10.9 1
13.2 1
13.3 1
14.3 1
> price.rel=price.freq/nrow(price.freq)
> price.rel
price
6.6 8.2
8.8 9.5 10.7 10.8 10.9 13.2 13.3
14.3
0.1 0.1
0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1
> barplot(price.freq)
> pie(price.freq)
> cbind(price.freq,price.rel,price.per)
>
* Cumulative frequency
Q.1
> duration = painters$Drawing
> duration
[1] 8 16 13 16 15 16 17 16 12 18 13 15 15 14 14
15 15 14 10 12 15 15 8 6 9
[26] 8 9 6 14
14 15 10 14 6 13 17 10 13 17 10 10
15 6 10
8 14 6 13 12 10
[51] 8 16 15 17
> range(duration)
[1] 6 18
> breaks = seq(5,20,by=5)
> breaks
[1] 5 10 15 20
> duration.cut = cut(duration,breaks)
> duration.cut
[1] (5,10] (15,20] (10,15] (15,20] (10,15] (15,20]
(15,20] (15,20] (10,15]
[10] (15,20] (10,15] (10,15] (10,15] (10,15] (10,15]
(10,15] (10,15] (10,15]
[19] (5,10]
(10,15] (10,15] (10,15] (5,10]
(5,10] (5,10] (5,10]
(5,10]
[28] (5,10]
(10,15] (10,15] (10,15] (5,10]
(10,15] (5,10] (10,15] (15,20]
[37] (5,10]
(10,15] (15,20] (5,10]
(5,10] (10,15] (5,10] (5,10]
(5,10]
[46] (10,15] (5,10]
(10,15] (10,15] (5,10]
(5,10] (15,20] (10,15] (15,20]
Levels: (5,10] (10,15] (15,20]
> duration.freq=table(duration.cut)
> duration.freq
duration.cut
(5,10] (10,15]
(15,20]
19 25
10
> duration.rel=duration.freq/nrow(painters)
> cbind(duration.freq,duration.rel)
duration.freq duration.rel
(5,10]
19 0.3518519
(10,15]
25 0.4629630
(15,20] 10
0.1851852
>
> duration.com = cumsum(duration.freq)
> cbind(duration.freq,duration.rel,duration.com)
duration.freq duration.rel duration.com
(5,10]
19 0.3518519 19
(10,15]
25 0.4629630 44
(15,20]
10 0.1851852 54
>
> ogive = c(0,duration.com)
> ogive
(5,10]
(10,15] (15,20]
0 19
44 54
> plot(ogive)
>
> plot(duration.freq)
Q-2
> library(MASS)
> data()
> cars
speed dist
1 4 2
2 4 10
3 7 4
4 7 22
5 8 16
6 9 10
7 10 18
8 10 26
9 10 34
10 11 17
11 11 28
12 12 14
13 12 20
14 12 24
15 12 28
16 13 26
17 13 34
18 13 34
19 13 46
20 14 26
21 14 36
22 14 60
23 14 80
24 15 20
25 15 26
26 15 54
27 16 32
28 16 40
29 17 32
30 17 40
31 17 50
32 18 42
33 18 56
34 18 76
35 18 84
36 19 36
37 19 46
38 19 68
39 20 32
40 20 48
41 20 52
42 20 56
43 20 64
44 22 66
45 23 54
46 24 70
47 24 92
48 24 93
49 24 120
50 25 85
> val=cars$speed
> val
[1] 4
4 7 7 8 9 10 10 10 11 11 12 12 12 12 13 13 13 13 14
14 14 14 15 15 15 16 16 17
[30] 17 17 18 18 18 18 19 19 19 20 20 20 20 20 22 23
24 24 24 24 25
> range(val)
[1] 4 25
> seq()
[1] 1
> breaks=seq(5,30,by=10)
> breaks
[1] 5 15 25
> val.cut=cut(val,breaks)
> val
[1] 4
4 7 7 8 9 10 10 10 11 11 12 12 12 12 13 13 13 13 14
14 14 14 15 15 15 16 16 17
[30] 17 17 18 18 18 18 19 19 19 20 20 20 20 20 22 23
24 24 24 24 25
> val.freq=table(val)
> val
[1] 4
4 7 7 8 9 10 10 10 11 11 12 12 12 12 13 13 13 13 14
14 14 14 15 15 15 16 16 17
[30] 17 17 18 18 18 18 19 19 19 20 20 20 20 20 22 23
24 24 24 24 25
> val.rel=val.freq/nrow(cars)
> val
[1] 4
4 7 7 8 9 10 10 10 11 11 12 12 12 12 13 13 13 13 14
14 14 14 15 15 15 16 16 17
[30] 17 17 18 18 18 18 19 19 19 20 20 20 20 20 22 23
24 24 24 24 25
> cbind(val.rel)
val.rel
4 0.04
7 0.04
8 0.02
9 0.02
10 0.06
11 0.04
12 0.08
13 0.08
14 0.08
15 0.06
16 0.04
17 0.06
18 0.08
19 0.06
20 0.10
22 0.02
23 0.02
24 0.08
25 0.02
> val.com=cumsum(val.freq)
> val
[1] 4
4 7 7 8 9 10 10 10 11 11 12 12 12 12 13 13 13 13 14
14 14 14 15 15 15 16 16 17
[30] 17 17 18 18 18 18 19 19 19 20 20 20 20 20 22 23
24 24 24 24 25
> val.com=cumsum(val.freq)
> val.com
4 7
8 9 10 11 12 13 14 15 16 17 18 19
20 22 23 24 25
2 4
5 6 9 11 15 19 23 26 28 31 35 38 43 44 45 49 50
> dis=cbind(val.freq,val.rel,val.com)
> dis
val.freq
val.rel val.com
4 2 0.04
2
7 2 0.04
4
8 1 0.02
5
9 1 0.02
6
10 3 0.06
9
11 2 0.04
11
12 4 0.08
15
13 4 0.08
19
14 4 0.08
23
15 3 0.06
26
16 2 0.04
28
17 3 0.06
31
18 4 0.08
35
19 3 0.06
38
20 5 0.10
43
22 1 0.02
44
23 1 0.02
45
24 4 0.08
49
25 1 0.02
50
> ogive=c(0,val.rel,breaks)
> ogive
4 7
8 9 10
11 12 13 14
15 16 17
18 19
0.00 0.04
0.04 0.02 0.02
0.06 0.04 0.08
0.08 0.08 0.06
0.04 0.06 0.08
0.06
20 22
23 24 25
0.10 0.02
0.02 0.08 0.02
5.00 15.00 25.00
> plot(ogive)
Q-3
> npk
> stat=npk$yield
> range(stat)
[1] 44.2 69.5
> seq()
[1] 1
> breaks=seq(5,10,by=5)
> breaks
[1] 5 10
> stat.cut=cut(stat,breaks)
> stat.cut
[1] <NA>
<NA> <NA> <NA> <NA> <NA> <NA> <NA>
<NA> <NA> <NA> <NA> <NA> <NA> <NA>
[16] <NA> <NA> <NA> <NA>
<NA> <NA> <NA> <NA> <NA>
Levels: (5,10]
> stat.freq=table(stat)
> stat.freq
stat
44.2 45.5 46.8 48.8 49.5 49.8 51.5 52 53.2
55 55.5 55.8 56 57 57.2 58.5
1 1
1 2 1
1 1 1
1 1 1
1 2 1
1 1
59 59.8 62 62.8 69.5
1
1 1 2
1
> stat.rel=stat.freq/nrow(npk)
> stat.rel
stat
44.2 45.5 46.8 48.8 49.5 49.8 51.5
0.04166667 0.04166667 0.04166667 0.08333333 0.04166667
0.04166667 0.04166667
52 53.2 55
55.5 55.8 56 57
0.04166667 0.04166667 0.04166667 0.04166667 0.04166667
0.08333333 0.04166667
57.2 58.5 59 59.8 62 62.8 69.5
0.04166667 0.04166667 0.04166667 0.04166667 0.04166667
0.08333333 0.04166667
> cbind(stat.rel)
stat.rel
44.2 0.04166667
45.5 0.04166667
46.8 0.04166667
48.8 0.08333333
49.5 0.04166667
49.8 0.04166667
51.5 0.04166667
52 0.04166667
53.2 0.04166667
55 0.04166667
55.5 0.04166667
55.8 0.04166667
56 0.08333333
57 0.04166667
57.2 0.04166667
58.5 0.04166667
59 0.04166667
59.8 0.04166667
62 0.04166667
62.8 0.08333333
69.5 0.04166667
> stat.com=cumsum(stat.freq)
> stat.com
44.2 45.5 46.8 48.8 49.5 49.8 51.5 52 53.2
55 55.5 55.8 56 57 57.2 58.5
1 2
3 5 6
7 8 9
10 11 12
13 15 16
17 18
59 59.8 62 62.8 69.5
19 20
21 23 24
> dis=cbind(stat.freq,stat.rel,stat.com)
> dis
stat.freq stat.rel stat.com
44.2 1
0.04166667 1
45.5 1
0.04166667 2
46.8 1
0.04166667 3
48.8 2
0.08333333 5
49.5 1
0.04166667 6
49.8 1
0.04166667 7
51.5 1
0.04166667 8
52 1
0.04166667 9
53.2 1 0.04166667 10
55 1
0.04166667 11
55.5 1
0.04166667 12
55.8 1
0.04166667 13
56 2
0.08333333 15
57 1
0.04166667 16
57.2 1
0.04166667 17
58.5 1
0.04166667 18
59 1
0.04166667 19
59.8 1
0.04166667 20
62 1
0.04166667 21
62.8 2
0.08333333 23
69.5 1
0.04166667 24
> ogive=c(0,stat.rel,breaks)
> ogive
44.2 45.5 46.8 48.8 49.5
0.00000000 0.04166667
0.04166667 0.04166667 0.08333333
0.04166667
49.8 51.5 52 53.2 55 55.5
0.04166667 0.04166667
0.04166667 0.04166667 0.04166667
0.04166667
55.8 56 57 57.2 58.5 59
0.04166667 0.08333333
0.04166667 0.04166667 0.04166667
0.04166667
59.8 62 62.8 69.5
0.04166667 0.04166667
0.08333333 0.04166667 5.00000000 10.00000000
> plot(ogive)
Measures of central
tendency and variability
Q-1
> faithful
> var=faithful$waiting
> var
[1] 79 54 74
62 85 55 88 85 51 85 54 84 78 47 83 52 62 84 52 79 51 47 78 69 74
[26] 83 55 76
78 79 73 77 66 80 74 52 48 80 59 90 80 58 84 58 73 83 64 53 82 59
[51] 75 90 54
80 54 83 71 64 77 81 59 84 48 82 60 92 78 78 65 73 82 56 79 71 62
[76] 76 60 78
76 83 75 82 70 65 73 88 76 80 48 86 60 90 50 78 63 72 84 75 51 82
[101] 62 88 49 83 81 47 84 52 86 81 75 59 89 79 59 81
50 85 59 87 53 69 77 56 88
[126] 81 45 82 55 90 45 83 56 89 46 82 51 86 53 79 81
60 82 77 76 59 80 49 96 53
[151] 77 77 65 81 71 70 81 93 53 89 45 86 58 78 66 76
63 88 52 93 49 57 77 68 81
[176] 81 73 50 85 74 55 77 83 83 51 78 84 46 83 55 81
57 76 84 77 81 87 77 51 78
[201] 60 82 91 53 78 46 77 84 49 83 71 80 49 75 64 76
53 94 55 76 50 82 54 75 78
[226] 79 78 78 70 79 70 54 86 50 90 54 54 77 79 64 75
47 86 63 85 82 57 82 67 74
[251] 54 83 73 73 88 80 71 83 56 79 78 84 58 83 43 60
75 81 46 90 46 74
> mean(var)
[1] 70.89706
> median(var)
[1] 76
> sd(var)
[1] 13.59497
> var(var)
[1] 184.8233
> range(var)
[1] 43 96
> max(var)
[1] 96
> min(var)
[1] 43
> range=max(var)-min(var)
> range
[1] 53
> summary(var)
Min. 1st
Qu. Median Mean 3rd Qu. Max.
43.0 58.0
76.0 70.9 82.0
96.0
> IQR(var)
[1] 24
> q1=quantile(var,.25)
> q1
25%
58
> q3=quantile(var,.75)
> q3
75%
82
> per32=quantile(var,.32)
> per32
32%
62.72
> cv=(sd(var)/mean(var)*100)
> cv
[1] 19.17565
> bxlt=boxplot(var)
> stem(var)
The decimal
point is 1 digit(s) to the right of the |
4 | 3
4 |
55566666777788899999
5 |
00000111111222223333333444444444
5 |
555555666677788889999999
6 |
00000022223334444
6 | 555667899
7 |
00001111123333333444444
7 |
555555556666666667777777777778888888888888889999999999
8 |
000000001111111111111222222222222333333333333334444444444
8 |
55555566666677888888999
9 |
00000012334
9 | 6
Q-2
> ships
> st=ship$service
Error: object 'ship' not found
> st=ship$Year
Error: object 'ship' not found
> data()
> BOD
Time demand
1 1 8.3
2 2 10.3
3 3 19.0
4 4 16.0
5 5 15.6
6 7 19.8
> var1=BOD$demand
> var1
[1] 8.3 10.3
19.0 16.0 15.6 19.8
> mean(var1)
[1] 14.83333
> meadian(var1)
Error in meadian(var1) : could not find function
"meadian"
> median(var1)
[1] 15.8
> sd(var1)
[1] 4.630623
> var(var1)
[1] 21.44267
> range(var1)
[1] 8.3 19.8
> min(var1)
[1] 8.3
> range=max(var1)-min(var1)
> range
[1] 11.5
> summary(var1)
Min. 1st Qu. Median
Mean 3rd Qu. Max.
8.30 11.62
15.80 14.83 18.25
19.80
> IQR(var1)
[1] 6.625
> qw=quantile(var1,.25)
> qw
25%
11.625
> qw1=quantile(var1,.75)
> qw1
75%
18.25
> per=quantile(var,.32)
Error in quantile.default(var, 0.32) :
anyNA()
applied to non-(list or vector) of type 'closure'
> per=quantile(var1,.32)
> per
32%
13.48
> cv=(sd(var1)/mean(var1)*100)
> cv
[1] 31.21768
> bxlt=boxplot(var1)
> stem(var1)
The decimal
point is 1 digit(s) to the right of the |
0 | 8
1 | 0
1 | 669
2 | 0
>
Q-3
> var1=coop$Conc
> var1
[1] 0.29 0.33
0.33 0.32 0.34 0.31 0.13 0.14 0.16 0.11 0.14 0.13 0.68 0.71 0.86 0.76 0.72
[18] 0.73 0.50
0.47 0.64 0.53 0.49 0.50 6.60 7.08 7.56 7.14 7.19 6.64 1.36 1.34 1.51 1.44
[35] 1.52 1.30 1.06
0.88 1.01 1.09 0.87 1.02 0.40 0.40 0.43 0.36 0.42 0.40 0.22 0.22 0.24
[52] 0.19 0.21
0.22 1.03 1.05 1.05 1.16 0.97 1.12 0.81 0.57 0.67 0.60 0.66 0.80 8.40 8.60
[69] 7.54 8.50
8.74 8.22 2.12 2.44 1.49 1.70 1.73 1.64 1.50 1.07 1.23 0.93 1.32 1.25 0.40
[86] 0.35 0.38
0.32 0.38 0.33 0.25 0.20 0.16 0.16 0.25 0.18 0.83 0.66 0.89 0.92 0.75 0.75
[103] 0.50 0.50 0.58 0.67 0.50 0.50 6.90 6.70 7.10
7.10 6.30 6.70 1.50 1.50 1.50 1.60 1.30
[120] 1.50 1.00 1.00 1.10 1.10 0.75 0.90 0.90 1.30
0.90 1.10 0.90 0.90 1.70 1.30 1.50 1.50
[137] 0.60 0.40 1.30 1.70 1.50 1.50 2.10 1.90 1.30
1.30 1.30 1.10 0.20 0.20 7.60 7.80 7.90
[154] 7.50 7.80 7.40 2.40 2.60 2.20 2.80 2.20 1.70
1.50 1.90 2.40 1.90 1.70 2.00 0.44 0.44
[171] 0.45 0.45 0.42 0.46 0.23 0.24 0.22 0.23 0.24
0.18 1.10 1.00 1.10 1.20 1.20 1.10 0.48
[188] 0.47 0.57 0.56 0.55 0.55 8.00 7.80 8.00 7.90
8.30 8.30 1.90 1.90 1.90 1.90 1.80 1.90
[205] 1.30 1.40 1.30 1.30 1.20 1.20 0.38 0.39 0.40
0.46 0.72 0.79 0.24 0.20 0.15 0.16 0.35
[222] 0.42 1.03 0.88 1.06 1.16 1.10 1.20 0.31 0.31
0.69 0.73 1.10 0.90 7.70 6.90 9.40 9.90
[239] 9.20 9.00 1.70 1.80 2.00 2.10 1.50 1.50 1.50
1.50 1.30 1.40 1.50 1.80
> mean(var1)
[1] 1.921508
> median(var1)
[1] 1.06
> sd(var1)
[1] 2.473207
> var(var1)
[1] 6.116755
> range(var1)
[1] 0.11 9.90
> max(var1)
[1] 9.9
> min(var2)
Error: object 'var2' not found
> range=max(var1)-min(var1)
> range
[1] 9.79
> summary(var1)
Min. 1st
Qu. Median Mean 3rd Qu. Max.
0.1100 0.4675
1.0600 1.9215 1.7000
9.9000
> IQR(var1)
[1] 1.2325
> c1=quantile(var1,.25)
> c1
25%
0.4675
> c2=quantile(var1,.75)
> c2
75%
1.7
> per1=quantile(var1,.32)
> per1
32%
0.6
> cv=(sd(var1)/mean(var1)*100)
> cv
[1] 128.7118
> bxlt=boxplot(var1)
> stem(var1)
The decimal
point is at the |
0 |
11111222222222222222222222223333333333334444444444444444444
0 |
55555555555555566666666677777777777888888889999999999999
1 |
000000001111111111111111222222223333333333333334444
1 |
55555555555555555566777777788899999999
2 | 0011122444
2 | 68
3 |
3 |
4 |
4 |
5 |
5 |
6 | 3
6 | 667799
7 | 111124
7 | 5566788899
8 | 002334
8 | 567
9 | 024
9 | 9
>
Binomial distribution
Q-1
> n=20
> x=5
> p=0.5
> ans=choose(n,x)*p^x*(1-p)^(n-x)
> ans
[1] 0.01478577
> dbinom(5,20,.5)
[1] 0.01478577
> ans=choose(20,5)
> ans
[1] 15504
>
ans=(choose(20,0)*.5^0*.5^20)+(choose(20,1)*.5^1*.5^19)+(choose(20,2)*.5^2*.5^18)
> ans
[1] 0.0002012253
>
ans=dbinom(0,20,.5)+dbinom(1,20,.5)+dbinom(2,20,.5)
> ans
[1] 0.0002012253
> ans=pbinom(2,20,.5)
> ans
[1] 0.0002012253
> ans=dbinom(2,20,.5)
> ans
[1] 0.0001811981
> ans=qbinom(0.0001811981,20,.5)
> ans
[1] 2
Q-2
> n=10
> x=6
> p=0.65
> ans=choose(n,x)*p^x*(1-p)^(n-x)
> ans
[1] 0.2376685
> dbinom(6,10,.65)
[1] 0.2376685
> ans=choose(10,6)
> ans
[1] 210
> pbinom(2,10,.65)
[1] 0.004821265
> qbinom(0.2376685
+ ,10,.65)
[1] 5
Q-3
> n=15
> x=8
> p=0.80
> ans=choose(n,x)*p^x*(1-p)^(n-x)
> ans
[1] 0.01381906
> dbinom(8,15,.80)
[1] 0.01381906
> pbinom(8,15,.80)
[1] 0.01805881
> qbinom( 0.01381906
+ ,15,.80)
[1] 8
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