STATISTIKA ANATASIA 125040200111140 Kelas : T

STATISTIKA

ANATASIA 125040200111140

Kelas : T

STATISTIKA

1. Banyaknya buah pisang yang tersengat hama dari 16 tanaman adalah 4, 9, 0, 1, 3, 24, 12, 3, 30, 12, 7, 13, 18, 4, 5, dan 15. Dengan menganggap data tersebut sebagai contoh, hitunglah varian, simpangan baku dan koefisien keragamannya. Statistik mana yang paling tepat untuk menggambarkan keragaman data tersebut?

Jawaban :

Untuk mengetahui simpangan baku dan varian, dicari dulu nilai rata2 data : μ=4 +9+0+…+5+15

16 =160

16 =10

Varian : σ²=

∑

i=1 N

(X_i−μ)² N

= 1088

16 = 68

Simpangan baku (standar deviasi):

σ =

√ ^∑

^{i=1N (XNi−μ)2 =}

^√

^{108816 = 8,246}

Koefisien keragaman : σ

μ x 100% = 8,246

10 x 100% = 82,46%

Statistik yang paling tepat untuk menggambarkan keragaman data tersebut adalah simpangan baku karena merupakan ukuran jauh dekatnya nilai pengamatan dari rata-rata hitungnya.

2. To study how first-grade students utilize their time when assigned to a math task, researcher observes 24 student and records their time off task out of 20 minutes. Times off task (minutes) : 4, 0, 2, 2, 4, 1, 4, 6, 9, 7, 2, 7, 5, 4,13, 7, 7, 10, 10, 0, 5, 3, 9 and 8. For this data set, find :

a) Mean and standard deviation, media and range Mean = (4+0+2+…+9+8)/24 = 129/224 = 5,375

Standar deviasi (rumus sama ky’ nmer 1) =

√

^{269,62524 = 3,35}

Median= data yang ditengah setelah diurutkan data ke 12 dan 13 yaitu sama-sama 5 jadi

Median = (5+5)/2 = 5

Range = nilai maksimum – nilai minimum = 13-0 = 13

b) Disply the data in the histogram plot, dot diagram and also stem-and-leaf diagram HISTOGRAM

C1

Frequency

12 10

8 6

4 2

0 5

4

3

2

1

0

Histogram of C1

DOT DIAGRAM

C1

12 10

8 6

4 2

0

Dotplot of C1

STEM-AND-LEAF DIAGRAM

Stem-and-leaf N = 24 Leaf Unit = 0.10 2 0 00

3 1 0 6 2 000 7 3 0 11 4 0000 (2) 5 00 11 6 0 10 7 0000 6 8 0 5 9 00 3 10 00 1 11 1 12 1 13 0

c) Determine the intervals `x ± s, `x ± 2s, `x ± 3s s = 3,35

´x ± s = 5,375 ± 3,35

´x ± 2s = 5,375 ± 6,7

´x ± 3s = 5,375 ± 10.05

d) Find the proportion of the meausurements that lie in each of this intervals.

e) Compare your finding with empirical guideline of bell-shaped distribution 3. The data below were obtained from the detailed record of purchases over several month. The

usage vegetables (in weeks) for a household taken from consumer panel were (gram) :

84 58 62 65 75 76 56 87 68 77 87 55 65 66 76 78 74 81 83 78 75 74 60 50 86 80 81 78 74 87

a. Plot a histogram of the data!

C2

Frequency

85 80 75 70 65

60 55

50 9

8 7 6 5 4 3 2 1 0

Histogram of C2

b. Find the relative frequency = fi/n of the usage time that did not exceed 80.

c. Calculate the mean, variance and the standard deviation Mean = 73,2

Variance = 108,0276 Standar deviasi = 10,39

d. Calculate the median and quartiles.

Median (nilai tengah) = 75,50 Quartil, ada Q1, Q2 (median) dan Q3

Qi = nilai yang ke (i(n+1))/4, setelah data diurutkan

Q1 = nilai yang ke (1*(30 + 1))/4 = data ke 7,75, jadi data ke 7 & 8 dijumlah dibagi 2 Hasilnya (65+65)/2 ---> Q1 = 65

Q3 = nilai yang ke (3*(30 + 1))/4 = data ke 23,25, jadi data ke 23 & 24 dijumlah dibagi 2 Hasilnya (81+81)/2 ---> Q3 = 81

4. The mean of corn weight is 278 g by ear and deviation standard is 9,64 g, and than we have 10 ears. If they are gotten from ten different fields, mean of plant height is Rp. 1200,- and its deviation standard is Rp 90,-, which one have more homogenous, the weight of corn ear or the

plant height? Explain your answer! Verify your results by direct calculation with the other data.

a. weight of corn ear

Mean = 278 ; standar deviasi = 9,64 ; N = 10 b. plant height

Mean = 1200 ; stndar deviasi = 90 ; N = 10

Dari data di atas, yang lebih homogen adalah data tinggi tanaman (plant height) karena jarak antara nilai (stndar deviasi x N) dengan nilai rata-rata (mean) lebih dekat dibandingkan data pada berat jagung (corn weight), dimana data yang homogen adalah jarak data pengamatan ke rata-rata relatif kecil atau dapat dikatakan datanya hampir seragam.

a. weight of corn ear

standar deviasi x N = 9,64 x 10 = 96,4 ; mean = 278 b. plant height

standar deviasi x N = 900 x 10 = 900 ; mean = 1200

jaraknya 900 ke 1200 lbih deket daripada jaraknya 96,4 ke 278 Contoh perhitungan data :

No Data A No Data B

1 13 1 30

2 11 2 12

3 12 3 24

4 10 4 5

5 11 5 33

6 13 6 9

7 14 7 17

8 12 8 61

9 10 9 46

10 11 10 15

Mean = 11,7 Mean = 25,20

Standar deviasi = 1,337 Standar deviasi = 17,69 ; N=10

5. The employment’s salary at seed company, abbreviated, as follows : 18, 15, 21, 19, 13, 15, 14, 23, 18 and 16 rupiah. If these abbreviation is real salary divide Rp. 100.000,-, find the mean, variance and standart deviasi!

Mean :

(18 + 15 + …..+ 16)/10 = 172/10 = 17.2  real salary (gaji sebenarnya) = 17.2 x Rp.100.000

= Rp. 1.720.000 Varian : 10,18 x 100.000 = 1.018.000

Standar deviasi : 3,19 x 100.000 = 319.000

6. Computer-aided statistical calculations. Calculation of the descriptive statistic such as `x and s are increasingly tedious with large data sets. Modern computers have come a long way in alleviating the drudgery of hand calculation. Microsoft Exel, Minitab or SPSS are three of computing packages those are easy accessible to student because its commands are in simple English. Find these programs and install its at your computers. Bellow main and sub menu of Microsoft Exel, Minitab and SPSS program. Use these software to find `x, s, s², and coefisien of variation (CV) for data set in exercise b. Histogram and another illustration can also be created.

Contoh data :

No Data

1 30

2 12

3 24

4 18

5 33

6 19

7 17

8 26

9 13

10 35

11 21

12 23

13 14

14 28

15 31

Descriptive Statistics

Mean StDev Variance CoefVar Q1 Median Q3 22.93 7.44 55.35 32.44 17.00 23.00 30.00

C2

Frequency

35 30

25 20

15 2.0

1.5

1.0

0.5

0.0

Histogram of C2

C2

33 30

27 24

21 18

15 12

Dotplot of C2

7. Some properties of the standard deviation

a) if a fixed number c is added to all measurements in a data set, will the deviations (xi -`x) remain changed? And consequently, will s² and s remain changed, too? Take data sample.

Misal c = 5, datanya sebagai berikut :

No

n (XA)

n+5 (XB)

(xA-

x)

(xB-

x)

1 13 18 -3.1 -3.1

2 16 21 -0.1 -0.1

3 17 22 0.9 0.9

4 15 20 -1.1 -1.1

5 10 15 -6.1 -6.1

6 21 26 4.9 4.9

7 23 28 6.9 6.9

8 11 16 -5.1 -5.1

9 15 20 -1.1 -1.1

10 20 25 3.9 3.9

mean (Xbar) 16.1 21.1

stndar deviasi (s) 4.25 4.25 varian (s² ) 18.10 18.10

Dari hasil perhitungan data di atas, nilai (xi -`x), s² dan s tetap sama meskipun ada penambahan angka c yaitu 5 pada data.

b) If all measurements in a data set are multiplied by a fixed number d, the deviation (xi -`x) get multiplied by d. Is it right? What about the s² and s? Take data sample.

Misal d = 3, datanya sebagai berikut :

No n

(XA) nx3

(XB) (xA- x) (xB-

x)

1 13 39 -3.1 -9.3

2 16 48 -0.1 -0.3

3 17 51 0.9 2.7

4 15 45 -1.1 -3.3

5 10 30 -6.1 -18.3

6 21 63 4.9 14.7

7 23 69 6.9 20.7

8 11 33 -5.1 -15.3

9 15 45 -1.1 -3.3

10 20 60 3.9 11.7

mean (Xbar) 16.1 48.3

stndar deviasi (s) 4.25 12.76 varian (s² ) 18.10 162.90

Benar, karena dilihat dari hasil perhitungan data di atas, apabila data dikalikan 3 maka nilai stndar deviasi (xi -`x) juga akan bernilai kali 3. Begitu pula untuk nilai s² dan s hasilnya sama jika nilainya lansung dikali 3, atau dihitung setelah datanya dikali 3.

c) Apply your computer software to explain your data sample. Verify your results by other data.