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Filledノート
䠄䝃䜲䝈
ఙ
(18)
ି ௫ି௑
௦
1. ㄗᕪㄽ
ఙ
ି ௫ି௬ మ Ȁଶ௦ మ
(19)
௦
┿್䛾᥎ᐃ್
మ Ȁଶ௦ మ
䛾ᶆᮏ䠅ㄗᕪ䛾☜⋡ᐦᗘ㛵ᩘ
䛜༑ศ኱䛝䛡䜜䜀䠈
ିఌమ Ȁଶఙ మ
䠄ẕ㞟ᅋ䠅ㄗᕪ䛾☜⋡ᐦᗘ㛵ᩘ
▱䜚䛯䛔䛾䛿ẕᩘ䠈䜟䛛䜛䛾䛿⤫ィ㔞
1. ㄗᕪㄽ
䜈䛾㔜䛽᭩䛝䛻䛴䛔䛶ㄝ᫂䛷䛝䜛䚹
2. ㄗᕪ䛾ศᕸ䠄ṇつศᕸ䠅䠈ㄗᕪㄽ䛸䛾ᑐᛂ䠈䝠䝇䝖䜾䝷䝮
ᯝ䛾⾲䛧᪉䛻䛴䛔䛶ㄝ᫂䛷䛝䜛䚹
ᕪཬ䜃䠄ᶆᮏ䠅ᶆ‽ㄗᕪ䠈᭷ຠᩘᏐ䠈┤᥋ ᐃ䛾 ᐃ⤖
1. ᐃ್䛸ㄗᕪ䠈ㄗᕪ䛾ἲ๎䠈᭱☜್䛸䠄ᶆᮏ䠅ᶆ‽೫
Ꮫ⩦┠ᶆ
⮬↛⛉Ꮫᐇ㦂ἲ
3
)
ଵ
ଶ
௡
┿್
䛾᥎ᐃ್
ଵ
1. ㄗᕪㄽ
‫ݕ‬
ଶ
݂௦ ‫ݔ ݕ‬ଵ ǡ ‫ݔ‬ଶ ൌ ݂௦ ‫ݔ ݕ‬ଵ ݂௦ ‫ݔ ݕ‬ଶ
(20)
್༙ᖜ䛜ᑠ
䛥䛟䛺䜛
⢭ᗘ䛜
ୖ䛜䜛
ᐃ್‫ݔ‬ଶ 䛻ᑐ䛧䛶‫ݕ‬䛜
ᐃ್ ‫ݔ‬ଵ 䛻ᑐ䛧䛶‫ݕ‬䛜 n = 2 䛾䛸䛝
┿್䛷䛒䜛☜⋡ᐦᗘ
┿್䛷䛒䜛☜⋡ᐦᗘ
ͳ
ͳ
మ
మ
మ
మ
݂௦ ሺ‫ݕ‬ȁ‫ݔ‬ଵ ሻ ൌ
݁ ି ௫భ ି௬ Ȁଶ௦
݂௦ ‫ݔ ݕ‬ଶ ൌ
݁ ି ௫ೞ ି௬ Ȁଶ௦
ʹߨ‫ݏ‬
ʹߨ‫ݏ‬
ᐃ䝕䞊䝍
ᐃ್䛻ᑐ䛧䛶┿್䛜 䛸䛺䜛☜⋡
1. ㄗᕪㄽ
⤫ィ㔞
ᶆᮏ䛻ᑐ䛧䛶ᖹᆒ(ᶆᮏᖹᆒ‫)ݔ‬䜔ศᩓ(୙೫ศᩓ ‫ ݏ‬ଶ )➼䜢ồ䜑䜛
݊ ൌᶆᮏ䛾኱䛝䛥(size) = ᶆᮏᩘ
• ᶆᮏ(sample): ẕ㞟ᅋ䛾㒊ศ㞟ྜ (ඖ䛾ᩘ
䛧䛛䛧䠈ᐇ㝿䛻඲䛶䛾ಶయ䛻ᑐ䛧䛶䝕䞊䝍䜢ྲྀ䜛䛾䛿ᅔ㞴
ẕᩘ
ẕ㞟ᅋ䛻ᑐ䛧䛶ᖹᆒ(ẕᖹᆒ ߤ)䜔ศᩓ(ẕศᩓߪ ଶ )➼䜢ồ䜑䛯䛔
ಶయ䛾ᩘ ܰ (ಶయᩘ) ౛䛘䜀 ඲10ṓඣ䛾㌟㛗
• ẕ㞟ᅋ(population): ㄪᰝᑐ㇟䛸䛺䜛඲䛶䛾ಶయ䛾㞟ྜ
ẕ㞟ᅋ䛸ᶆᮏ
4
2
ʹ‫ ݕ‬ଶ
଴
௜
(21)
଴
1. ㄗᕪㄽ
௑బ
௜ୀଵ
ଶ
௜
ᶆᮏᶆ‽ㄗᕪ ௠
䛾
1. ㄗᕪㄽ
ᶆᮏᶆ‽೫ᕪ
ᶆᮏᶆ‽ㄗᕪ ௠ 䛿
ಸ
ಶ䚻䛾 ᐃ್䛾䜀䜙䛴䛝ලྜ ᶆᮏᶆ‽೫ᕪ
᭱☜್䛾ᶆᮏᶆ‽೫ᕪ ௑బ
᭱☜್䛸ᶆᮏᶆ‽ㄗᕪ
68.3%䛾ಙ㢗ᗘ䛾⠊ᅖ:
䝕䞊䝍䛜2ಶ䛾䛸䛝
(22)
‫݂ ן‬௦೉బ ‫ܺ ݕ‬଴
(26)
(23)
᭱☜್ X0 䛾☜⋡ᐦᗘ
ଶ
7
5
ͳ
ʹ‫ ݕ‬ଶ െ ʹ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ‫ ݕ‬൅ ‫ݔ‬ଵଶ ൅ ‫ݔ‬ଶଶ
‡š’ െ
ଶ
ʹߨ‫ݏ‬
ʹ‫ ݏ‬ଶ
䛜᭱ᑠ䛻䛺䜛 ‫ ݕ‬䛾್䛜᭱☜್ ܺͲ
ൌ
᭱☜್䛿⟬⾡ᖹᆒ䛰䛳䛯䟿
൅ ‫ݔ‬ଶଶ
ଶ
‫ ݕ‬െ ܺ଴ ଶ
‫ ݕ‬െ ܺ଴
‫ ’š‡ ן‬െ
ൌ
‡š’
െ
ʹሺ‫ ݏ‬ଶ Ȁʹሻ
ʹ‫ݏ‬௑ଶబ
௜ୀଵ
ଶ
െ ʹ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ‫ ݕ‬൅
‫ݔ‬ଵଶ
ͳ
‫ݔ‬ଵ െ ‫ ݕ‬ଶ ൅ ‫ݔ‬ଶ െ ‫ݕ‬
‡š’ െ
ଶ
ʹߨ‫ݏ‬
ʹ‫ ݏ‬ଶ
݂௦ ‫ݔ ݕ‬ଵ ǡ ‫ݔ‬ଶ
ൌ
݂௦ ‫ݔ ݕ‬ଵ ǡ ‫ݔ‬ଶ ൌ ݂௦ ‫ݔ ݕ‬ଵ ݂௦ ሺ‫ݕ‬ȁ‫ݔ‬ଶ ሻ䛜᭱኱䛸䛺䜛 䜢䛂᭱☜್䛃䛸䛔䛖
ᐇ⌧䛧䛭䛖䛺 y 䛿䠛
௡
݊
‫ ݕ‬െ ܺ଴
௜ୀଵ
ͳ
෍ ‫ ݕ‬െ ‫ݔ‬௜
ʹ‫ ݏ‬ଶ
ൌ
ൌ
ʹߨ‫ݏ‬
ͳ
௡
௜ୀଵ
௡
௜ୀଵ
௡
‡š’ െ‫ݔ ݕ ܮ‬௜
䚷
௑బ
௠
௜ୀଵ
଴
෍ሺ‫ݔ‬௜ െ ܺ଴ ሻ
1. ㄗᕪㄽ
(24)
ʹ‫ ݏ‬ଶ
௜ୀଵ
௡
௜
ଶ
௜ୀଵ
௡
ଶ
௜
௜ୀଵ
௡
௜
௑బ
௠
1. ㄗᕪㄽ
௜ୀଵ
௡
௜
଴
ଶ
଴
ଶ
ଶ
(28)
(27)
(25)
௜ୀଵ
ᶆᮏᶆ‽೫ᕪ䛸ᶆᮏᶆ‽ㄗᕪ
2
୙೫ศᩓ = [ᶆᮏᶆ‽೫ᕪ]
ଶ
൅
ᶆᮏᶆ‽ㄗᕪ=᭱☜್䛾ᶆᮏᶆ‽೫ᕪ
ଶ
௑బ
ʹ‫ݏ‬௑ଶబ
௡
ͳ
෍ ‫ݔ‬௜
݊
ͳ
݊‫ ݕ‬ଶ െ ʹ‫ ݕ‬෍ ‫ݔ‬௜ ൅ ෍ ‫ݔ‬௜ଶ
ʹ‫ ݏ‬ଶ
௡
ൌ
ͳ
ଶ
ൌ
௡
ଶ
ଶ
݊̾䛾䝕䞊䝍 ൌ ʹ‫ ݏ‬ଶ ‫ ݕ‬െ ʹ‫ ݊ ݕ‬෍ ‫ݔ‬௞ ൅ ʹ‫ ݏ‬ଶ ෍ ‫ݔ‬௜
௜ୀଵ
௜ୀଵ
䛻ᑐ䛩䜛┿್
ଶ
௡
௡
ͳ
ͳ
ͳ
䛾᥎ᐃ್‫ݕ‬䛾 ൌ ͳ
‫ݕ‬
െ
෍
‫ݔ‬
൅
෍
‫ݔ‬௜ଶ െ
௜
ଶ
ଶ
ʹሺ‫ ݏ‬Ȁ݊ሻ
݊
ʹ‫ݏ‬
ʹ ‫ ݏ‬ଶ Ȁ݊
ศᕸ㛵ᩘ䛾
௜ୀଵ
௜ୀଵ
௡
ͳ
ͳ
ศᩓ
ଶ
ଶ
ଶ
‫ݔ ݕ ܮ‬௜
݂௦ ‫ݔ ݕ‬௜ ݅ ൌ ͳǡ ‫ ڮ‬ǡ ݊
ͳ
ಶ䛾䝕䞊䝍{xi}䛾ሙྜ
8
6
୫ୟ୶ =2.95
୫୧୬
=1.95
༊㛫ᩘ
ଵȀଶ
䛾༊㛫䛻ศ๭
ଶ
2. 䝠䝇䝖䜾䝷䝮䛸ṇつศᕸ᭤⥺
䛾㡿ᇦ䜢
or
9
11
䝇䝍䞊䝆䜵䝇䛾බᘧ
᭱ᑠ್䛸᭱኱್䜢ྵ䜐䜘䛖䛻䝕䞊䝍㡿ᇦ䜢Ỵ䜑䜛
䝕䞊䝍䛾㡿ᇦ
=min(䝕䞊䝍䛾ධ䛳䛶䛔䜛䝉䝹䛾⠊ᅖ)
=min(A1:T20) 2.02
᭱ᑠ್
=max(䝕䞊䝍䛾ධ䛳䛶䛔䜛䝉䝹䛾⠊ᅖ)
=max(A1:T20) 2.88
᭱኱್
᭱኱್䞉᭱ᑠ್䞉༊㛫ᖜ
2. 䝠䝇䝖䜾䝷䝮䛸ṇつศᕸ᭤⥺
–ṇつศᕸ᭤⥺䛾సᡂ
–䝠䝇䝖䜾䝷䝮䜢సᡂ (ᗘᩘศᕸ⾲䛾ྍど໬)
4.ྛ䝕䞊䝍༊㛫䛻ྵ䜎䜜䜛䝕䞊䝍䛾ᩘ݊௞ 䜢ᩘ䛘䜛
3.㝵⣭್(ྛ䝕䞊䝍༊㛫䛾୰ኸ್) ௞ 䜢Ỵ䜑䜛
z ྛ䝕䞊䝍༊㛫 ሾ‫ݔ‬௞ െ ȟ‫ݔ‬Ȁʹǡ ‫ݔ‬௞ ൅ ȟ‫ݔ‬Ȁʹሻ䠄݇ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܯ‬䠅
2.䝕䞊䝍༊㛫䛾ಶᩘ K, 䝕䞊䝍༊㛫䛾ᖜȟ‫ݔ‬䜢Ỵ䜑䜛
1.䝕䞊䝍䛾᭱኱್䠈᭱ᑠ್䠈᭱☜್䠈ᶆᮏᶆ‽೫ᕪ䠈ᶆᮏᶆ
‽ㄗᕪ䜢ồ䜑䜛
–ᗘᩘศᕸ⾲
ᡭ㡰
2. 䝠䝇䝖䜾䝷䝮䛸ṇつศᕸ᭤⥺
଴
௜ୀଵ
݉
௜ୀଵ
௡
௜
௜
଴
ଶ
2. 䝠䝇䝖䜾䝷䝮䛸ṇつศᕸ᭤⥺
=stdev(A1:J10)/sqrt(100) 0.0174
௠
=stdev(䝕䞊䝍䛾ධ䛳䛶䛔䜛䝉䝹䛾⠊ᅖ)/sqrt(䝕䞊䝍ᩘ)
ᶆᮏᶆ‽ㄗᕪ
=stdev(䝕䞊䝍䛾ධ䛳䛶䛔䜛䝉䝹䛾⠊ᅖ)
=stdev(A1:J10) 0.174
ᶆᮏᶆ‽೫ᕪ
଴
=ᖹᆒ್
=average(䝕䞊䝍䛾ධ䛳䛶䛔䜛䝉䝹䛾⠊ᅖ)
௡
=average(A1:J10) 2.46
᭱☜್
᭱☜್䞉ᶆᮏᶆ‽೫ᕪ䞉ᶆᮏᶆ‽ㄗᕪ
2. 䝠䝇䝖䜾䝷䝮䛸ṇつศᕸ᭤⥺
䝕䞊䝍
12
10
3
16
21
23
20
6
2
3
[ 2.15, 2.25 ) 2.20
[ 2.25, 2.35 ) 2.30
[ 2.35, 2.45 ) 2.40
[ 2.45, 2.55 ) 2.50
[ 2.55, 2.65 ) 2.60
[ 2.65, 2.75 ) 2.70
[ 2.75, 2.85 ) 2.80
[ 2.85, 2.95 ) 2.90
10
15
20
25
0
5
ܾ
௔
2. 䝠䝇䝖䜾䝷䝮䛸ṇつศᕸ᭤⥺
ଶ
ሾܽǡ ܾሻ䛾༊㛫䛻䛒䜛䝕䞊䝍䛾ᗘᩘ䛻ᑐᛂ
ሾ௔ǡ௕ሻ
௕
݊ᅇ䛾 ᐃ䛷ܽ䛛䜙ܾ䛾㛫䛻 ᐃ䛥䜜䜛䛸
ᮇᚅ䛥䜜䜛 ᐃ್䛾ᩘ ݊ොሾ௔ǡ௕ሻ
ᐃ್ ‫ ݔ‬䛜 ܽ䛛䜙 ܾ䛾㛫䛷䛒䜛☜⋡
ܽ
௞
2
2.2
2.4
2.8
3
‫ݏ‬ൌ
2. 䝠䝇䝖䜾䝷䝮䛸ṇつศᕸ᭤⥺
2.6
௞
଴
௡
௜ୀଵ
ͳ
෍ ‫ݔ‬௜ െ ܺ଴
݊െͳ
௜ୀଵ
௡
15
ଶ
: 䝕䞊䝍䛾⥲ᩘ
: ༊㛫ᖜ
௞ : 㝵⣭್
଴ : ᭱☜್
: ᶆᮏᶆ‽೫ᕪ
ଶ
ͳ
ܺ଴ ൌ ෍ ‫ݔ‬௜
݊
㝵⣭್㛫䛾 ‫ݔ‬䛻ᑐ䛧䛶䜒ᗘᩘ䛾ᮇᚅ್䜢ィ
⟬䛩䜛䛸䜘䛔䚹
ᗘᩘ䛾ᮇᚅ್
௞
13
௞
௞
௞
ଶ
଴
ெ
௜ୀଵ
䝠䝇䝖䜾䝷䝮䛾䝢䞊䜽್䛾༙ศ䛾್䛸䛺䜛Წ䜾䝷䝣䜢ぢ䛴䛡䛶䠈䛭䛾ᖜ
(ȟߝ ) 䛛䜙ᶆᮏᶆ‽೫ᕪ䜢᥎ᐃ䛩䜛䠄ȟߝ ؒ ʹǤ͵ͷ‫ݏ‬䠅
– 䝠䝇䝖䜾䝷䝮䛛䜙್༙ᖜ䜢౑䛳䛶
2. 䝠䝇䝖䜾䝷䝮䛸ṇつศᕸ᭤⥺
– ᭱☜್ ܺ଴ 䜢୰ᚰ䛻ܺ଴ േ ‫ݏ‬ሺܺ଴ േ ʹ‫ݏ‬ሻ䛾⠊ᅖ䛾䝕䞊䝍䛿඲
䝕䞊䝍䛾ఱ%䛛䠛
• 㠃✚๎䜢౑䛳䛶
14
16
ൌ ܿ ൅ ෍ ݊௜ ‫ݔ‬௜ െ ܿ Ȁሺ݊ଵ ൅ ‫ ڮ‬൅ ݊ெ ሻ
௜ୀଵ
‫ݔ‬ҧ ൌ ෍ ݊௜ ‫ݔ‬௜ Ȁ ݊ଵ ൅ ‫ ڮ‬൅ ݊ெ
䝕䞊䝍䜢ᑠ䛥䛔㡰䛻୪䜉䛶䠈䝕䞊䝍䛾ᩘ䛷4➼ศ䛧䚸2䞉3␒┠䛾༊㛫ᖜ
(ʹ‫ )ݎ‬䛛䜙ᶆᮏᶆ‽೫ᕪ䜢᥎ᐃ䛩䜛䠄ʹ‫ͳ ؒ ݎ‬Ǥ͵ͷ‫ݏ‬䠅
– 䠄䝕䞊䝍䛛䜙䠅1/4 㠃✚๎
• ᶆᮏᶆ‽೫ᕪ
Ȃ ݊௜ : 㝵⣭݅䛾ᗘᩘ䠈 ‫ݔ‬௜ : 㝵⣭್
䈄 ᶆᮏᖹᆒ 䠄ᗘᩘศᕸ⾲䛛䜙䠅ெ
䝕䞊䝍䞉ᗘᩘศᕸ⾲䞉䝠䝇䝖䜾䝷䝮䛛䜙
ଶ
䛾ᗘᩘ
௞ ᐃ䛥䜜䛯ᗘᩘ
2. 䝠䝇䝖䜾䝷䝮䛸ṇつศᕸ᭤⥺
௫ೖ ି୼௫Ȁଶ
௫ೖ ା୼௫Ȁଶ
ᗘᩘ䛾ᮇᚅ್
௞
༊㛫
ᗘᩘ
䝠䝇䝖䜾䝷䝮䛸ṇつศᕸ᭤⥺䜢㔜䛽䜛
ሾ‫ݔ‬௞ െ ȟ‫ݔ‬Ȁʹǡ ‫ݔ‬௞ ൅ ȟ‫ݔ‬Ȁʹሻ
5
[ 2.05, 2.15 ) 2.10
݊௞
1
‫ݔ‬௞
㝵⣭್ ᗘᩘ
[ 1.95, 2.05 ) 2.00
‫ݔ‬௞ േ ȟ‫ݔ‬Ȁʹ
㝵⣭
ᗘᩘ䛸☜⋡ศᕸ
ᗘᩘ
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