Jika Z berdistribusi normal dengan mean 0 dan variansi 1, untuk setiap nilai z, tabel memberikan nilai Φ(z), dengan Φ(z) = P(Z ≤ z).
Untuk nilai z negatif, gunakan Φ(−z) = 1 − Φ(z).
z | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
0,0 | 0,5000 | 0,5040 | 0,5080 | 0,5120 | 0,5160 | 0,5199 | 0,5239 | 0,5279 | 0,5319 | 0,5359 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 |
0,1 | 0,5398 | 0,5438 | 0,5478 | 0,5517 | 0,5557 | 0,5596 | 0,5636 | 0,5675 | 0,5714 | 0,5753 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 |
0,2 | 0,5793 | 0,5832 | 0,5871 | 0,5910 | 0,5948 | 0,5987 | 0,6026 | 0,6064 | 0,6103 | 0,6141 | 4 | 8 | 12 | 15 | 19 | 23 | 27 | 31 | 35 |
0,3 | 0,6179 | 0,6217 | 0,6255 | 0,6293 | 0,6331 | 0,6368 | 0,6406 | 0,6443 | 0,6480 | 0,6517 | 4 | 7 | 11 | 15 | 19 | 22 | 26 | 30 | 34 |
0,4 | 0,6554 | 0,6591 | 0,6628 | 0,664 | 0,6700 | 0,6736 | 0,6772 | 0,6808 | 0,6844 | 0,6879 | 4 | 7 | 11 | 14 | 18 | 22 | 25 | 29 | 32 |
0,5 | 0,6915 | 0,6950 | 0,6985 | 0,7019 | 0,7054 | 0,7088 | 0,7123 | 0,7157 | 0,7190 | 0,7224 | 3 | 7 | 10 | 14 | 17 | 20 | 24 | 27 | 31 |
0,6 | 0,7257 | 0,7291 | 0,7324 | 0,7357 | 0,7389 | 0,7422 | 0,7454 | 0,7486 | 0,7517 | 0,7549 | 3 | 7 | 10 | 13 | 16 | 19 | 23 | 26 | 29 |
0,7 | 0,7580 | 0,7611 | 0,7642 | 0,7673 | 0,7704 | 0,7734 | 0,7764 | 0,7794 | 0,7823 | 0,7852 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 |
0,8 | 0,7881 | 0,7910 | 0,7939 | 0,7967 | 0,7995 | 0,8023 | 0,8051 | 0,8078 | 0,8106 | 0,8133 | 3 | 5 | 8 | 11 | 14 | 16 | 19 | 22 | 25 |
0,9 | 0,8159 | 0,8186 | 0,8212 | 0,8238 | 0,8264 | 0,8289 | 0,8315 | 0,8340 | 0,8365 | 0,8389 | 3 | 5 | 8 | 10 | 13 | 15 | 18 | 20 | 23 |
1,0 | 0,8413 | 0,8438 | 0,8461 | 0,8485 | 0,8508 | 0,8531 | 0,8554 | 0,8577 | 0,8599 | 0,8621 | 2 | 5 | 7 | 9 | 12 | 14 | 16 | 19 | 21 |
1,1 | 0,8643 | 0,8665 | 0,8686 | 0,8708 | 0,8729 | 0,8749 | 0,8770 | 0,8790 | 0,8810 | 0,8830 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
1,2 | 0,8849 | 0,8869 | 0,8888 | 0,8907 | 0,8925 | 0,8944 | 0,8962 | 0,8980 | 0,8997 | 0,9015 | 2 | 4 | 6 | 7 | 9 | 11 | 13 | 15 | 17 |
1,3 | 0,9032 | 0,9049 | 0,9066 | 0,9082 | 0,9099 | 0,9115 | 0,9131 | 0,9147 | 0,9162 | 0,9177 | 2 | 3 | 5 | 6 | 8 | 10 | 11 | 13 | 14 |
1,4 | 0,9192 | 0,9207 | 0,9222 | 0,9236 | 0,9251 | 0,9265 | 0,9279 | 0,9292 | 0,9306 | 0,9319 | 1 | 3 | 4 | 6 | 7 | 8 | 10 | 11 | 13 |
1,5 | 0,9332 | 0,9345 | 0,9357 | 0,9370 | 0,9382 | 0,9394 | 0,9406 | 0,9418 | 0,9429 | 0,9441 | 1 | 2 | 4 | 5 | 6 | 7 | 8 | 10 | 11 |
1,6 | 0,9452 | 0,9463 | 0,9474 | 0,9484 | 0,9495 | 0,9505 | 0,9515 | 0,9525 | 0,9535 | 0,9545 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
1,7 | 0,9554 | 0,9564 | 0,9573 | 0,9582 | 0,9591 | 0,9599 | 0,9608 | 0,9616 | 0,9625 | 0,9633 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | 8 |
1,8 | 0,9641 | 0,9649 | 0,9656 | 0,9664 | 0,9671 | 0,9678 | 0,9686 | 0,9693 | 0,9699 | 0,9706 | 1 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 6 |
1,9 | 0,9713 | 0,9719 | 0,9726 | 0,9732 | 0,9738 | 0,9744 | 0,9750 | 0,9756 | 0,9761 | 0,9767 | 1 | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 |
2,0 | 0,9772 | 0,9778 | 0,9783 | 0,9788 | 0,9793 | 0,9798 | 0,9803 | 0,9808 | 0,9812 | 0,9817 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
2,1 | 0,9821 | 0,9826 | 0,9830 | 0,9834 | 0,9838 | 0,9842 | 0,9846 | 0,9850 | 0,9854 | 0,9857 | 0 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 4 |
2,2 | 0,9861 | 0,9864 | 0,9868 | 0,9871 | 0,9875 | 0,9878 | 0,9881 | 0,9884 | 0,9887 | 0,9890 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 |
2,3 | 0,9893 | 0,9896 | 0,9898 | 0,9901 | 0,9904 | 0,9906 | 0,9909 | 0,9911 | 0,9913 | 0,9916 | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
2,4 | 0,9918 | 0,9920 | 0,9922 | 0,9925 | 0,9927 | 0,9929 | 0,9931 | 0,9932 | 0,9934 | 0,9936 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
2,5 | 0,9938 | 0,9940 | 0,9941 | 0,9943 | 0,9945 | 0,9946 | 0,9948 | 0,9949 | 0,9951 | 0,9952 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
2,6 | 0,9953 | 0,9955 | 0,9956 | 0,9957 | 0,9959 | 0,9960 | 0,9961 | 0,9962 | 0,9963 | 0,9964 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
2,7 | 0,9965 | 0,9966 | 0,9967 | 0,9968 | 0,9969 | 0,9970 | 0,9971 | 0,9972 | 0,9973 | 0,9974 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
2,8 | 0,9974 | 0,9975 | 0,9976 | 0,9977 | 0,9977 | 0,9978 | 0,9979 | 0,9979 | 0,9980 | 0,9981 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
2,9 | 0,9981 | 0,9982 | 0,9982 | 0,9983 | 0,9984 | 0,9984 | 0,9985 | 0,9985 | 0,9986 | 0,9986 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
SOAL LATIHAN
Diketahui \(\textbf{Z} \sim \textbf{N}(0, 1)\)
Soal 01
\(\text{P(Z ≤ 1,23)}\)
Gunakan tabel distribusi normal standar
z | 3↓ |
1,2 → | 0,8907 |
\(\text{P(Z ≤ 1,23) = 0,8907}\)
Soal 02
\(\text{P(Z < 2,86)}\)
z | 6↓ |
2,8 → | 0,9979 |
Soal 03
\(\text{P(Z < −1,82)}\)
\(\text{P(Z <−1,82)} = 1\:-\: \text{P(Z < 1,82)}\)
Untuk \(\text{P(Z < 1,82)}\) nilainya adalah 0,9656
z | 2↓ |
1,8 → | 0,9656 |
\(\text{P(Z < −1,82)} = 1\:-\: \text{P(Z < 1,82)}\)
\(\text{P(Z < −1,82)} = 1\:-\: 0,9656\)
\(\text{P(Z < −1,82)} = 0,0344\)
Soal 04
\(\text{P(Z > 0,45)}\)
\(\text{P(Z > 0,45)} = 1\:-\: \text{P(Z < 0,45)} \)
Lihat tabel distribusi normal standar untuk nilai \(\text{P(Z < 0,45)}\).
z | 5↓ |
0,4 → | 0,6736 |
\(\text{P(Z > 0,45)} = 1\:-\: \text{P(Z < 0,45)} \)
\(\text{P(Z > 0,45)} = 1\:-\: 0,6736 \)
\(\text{P(Z > 0,45)} = 0,3264 \)
Soal 05
\(\text{P(Z ≥ −1,73)}\)
\(\text{P(Z ≥ −1,73)} = \text{P(Z ≤ 1,73)}\)
z | 3↓ |
1,7 → | 0,9582 |
\(\text{P(Z ≥ −1,73)} = \text{P(Z ≤ 1,73)} = 0,9582\)
Soal 06
\(\text{P(1,15 < Z < 1,35)}\)
\(\text{P(1,15 < Z < 1,35)} = \text{P(Z < 1,35)}\:-\: \text{P(Z < 1,15)}\)
Lihat tabel distribusi normal standar untuk nilai \(\text{P(Z < 1,35)}\).
z | 5↓ |
1,3 → | 0,9115 |
Lihat tabel distribusi normal standar untuk nilai \(\text{P(Z < 1,35)}\).
z | 5↓ |
1,1 → | 0,8749 |
\(\text{P(1,15 < Z < 1,35)} = \text{P(Z < 1,35)}\:-\: \text{P(Z < 1,15)}\)
\(\text{P(1,15 < Z < 1,35)} = 0,9115\:-\: 0,8749\)
\(\text{P(1,15 < Z < 1,35)} =0,0366\)
Soal 07
\(\text{P(−1,96 < Z < 1,96)}\)
\(\text{P(−1,96 < Z < 1,96)} = \text{P(Z < 1,96)} \:-\: \text{P(Z < −1,96)}\)
\(\text{P(−1,96 < Z < 1,96)} = \text{P(Z < 1,96)} \:-\: [1\:-\:\text{P(Z < 1,96)}]\)
Lihat tabel distribusi normal standar untuk nilai \(\text{P(Z < 1,96)}\).
z | 6↓ |
1,9 → | 0,9750 |
\(\text{P(−1,96 < Z < 1,96)} = \text{P(Z < 1,96)} \:-\: [1\:-\:\text{P(Z < 1,96)}]\)
\(\text{P(−1,96 < Z < 1,96)} = 0,9750 \:-\: [1\:-\:0,9750]\)
\(\text{P(−1,96 < Z < 1,96)} = 0,9750 \:-\: 0,025\)
\(\text{P(−1,96 < Z < 1,96)} = 0,95\)
Soal 08
\(\text{P(−1,52 < Z < −0,52)}\)
\(\text{P(−1,52 < Z < −0,52)} = \text{P(Z < −0,52)}\:-\:\text{P(Z < −1,52)}\)
\(\text{P(−1,52 < Z < −0,52)} = 1\:-\:\text{P(Z < 0,52)}\:-\:[1\:-\:\text{P(Z < 1,52)}]\)
\(\text{P(−1,52 < Z < −0,52)} = 1\:-\:\text{P(Z < 0,52)}\:-\:1 + \text{P(Z < 1,52)}\)
\(\text{P(−1,52 < Z < −0,52)} = -\text{P(Z < 0,52)}+ \text{P(Z < 1,52)}\)
\(\text{P(−1,52 < Z < −0,52)} = \text{P(Z < 1,52)} -\text{P(Z < 0,52)}\)
Lihat tabel distribusi normal standar untuk nilai \(\text{P(Z < 0,52)}\)
z | 2↓ |
0,5 → | 0,6985 |
Lihat tabel distribusi normal standar untuk nilai \(\text{P(Z < 1,52)}\)
z | 2↓ |
1,5 → | 0,9357 |
\(\text{P(−1,52 < Z < −0,52)} = \text{P(Z < 1,52)} \:-\:\text{P(Z <0,52)}\)
\(\text{P(−1,52 < Z < −0,52)} = 0,9357\:-\:0,6985\)
\(\text{P(−1,52 < Z < −0,52)} = 0,2372\)
Soal 09
\(\text{P( Z < 0,981)}\)
z | 8 ↓ | ADD 1 ↓ |
0,9 → | 0,8365 | 3 |
\(\text{P( Z < 0,981)}\) = 0,8365 + 0,0003 = 0,8368
Soal 10
\(\text{P( Z > −2,052)}\)
\(\text{P( Z > −2,052)} = \text{P( Z < 2,052)}\)
z | 5 ↓ | ADD 2 ↓ |
2,0 → | 0,9798 | 1 |
\(\text{P( Z > −2,052)} = \text{P( Z < 2,052)}\) = 0,9798 + 0,0001 = 0,9799
Diketahui \(\textbf{Z} \sim \textbf{N}(0, 1)\). Tentukan nilai \(a\) jika:
Soal 11
\(\text{P}(\text{Z} \leq a) = 0,9406\)
z | 6 |
1,5 | ← 0,9406 ↑ |
\(\text{P}(\text{Z} \leq 1,56) = 0,9406\)
Jadi, nilai \(a\) adalah 1,56
Soal 12
\(\text{P}(\text{Z} < a) = 0,7227\)
0,7227 berasal dari 0,7224 + 0,0003
z | 9 | 1 |
0,5 | ← 0,7224 ↑ | 3 ↑ |
\(\text{P}(\text{Z} < 0,591) = 0,7227\)
Jadi, nilai \(a\) adalah 0,591
Soal 13
\(\text{P}(\text{Z} > a) = 0,0232\)
\(1\:-\: \text{P}(\text{Z}< a) = 0,0232\)
\(\text{P}(\text{Z}< a) = 1\:-\:0,0232\)
\(\text{P}(\text{Z}< a) = 0,9768\)
0,9768 berasal dari 0,9767 + 0,001
z | 9 | 1 |
1,9 | ← 0,9767 ↑ | 1 ↑ |
\(\text{P}(\text{Z}< 1,991) = 0,9768\)
Jadi, nilai \(a\) adalah 1,991.
Soal 14
\(\text{P}(\text{Z}\geq -a) = 0,9673\)
\(\text{P}(\text{Z}\geq -a) = 0,9673\)
\(\text{P}(\text{Z}\leq a) = 0,9673\)
0,9673 berasal dari 0,9671 + 0,0002
z | 4 | 3 |
1,8 | ← 0,9671 ↑ | 2 ↑ |
\(\text{P}(\text{Z}\leq 1,843) = 0,9673\)
Jadi, nilai \(a\) adalah 0,9673.
Soal 15
\(\text{P}(-a < \text{Z}< a) = 0,8230\)
\(\text{P}(-a < \text{Z}< a) = 0,8230\)
\(\text{P}(\text{Z}< a)\:-\:\text{P}(\text{Z}< -a) = 0,8230\)
\(\text{P}(\text{Z}< a)\:-\:[1\:-\:\text{P}(\text{Z}<a)] = 0,8230\)
\(2\text{P}(\text{Z}< a)\:-\:1 = 0,8230\)
\(2\text{P}(\text{Z}< a) = 1,8230\)
\(\text{P}(\text{Z}< a) = 0,9115\)
z | 5 |
1,3 | ← 0,9115 ↑ |
\(\text{P}(\text{Z}< 1,35) = 0,9115\)
Jadi, nilai \(a\) adalah 1,35.