Опря А.Т. Статистика (модульний варіант з програмованою формою контролю знань)
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ɤɨɦɩɥɟɤɫɭ. ȼ ɚɧɚɥɿɡɿ ɟɤɨɧɨɦɿɱɧɢɯ ɹɜɢɳ ɧɚɣɱɚɫɬɿɲɟ ɡɭɫɬɪɿɱɚɸɬɶɫɹ
ɧɟɪɿɜɧɨɦɿɪɧɿ ɤɨɦɩɥɟɤɫɢ. Ⱦɥɹ ɧɢɯ ɧɚɛɢɪɚɽ ɫɢɥɢ ɧɟɪɿɜɧɿɫɬɶ
ɋ
Ⱥ
+ɋ
ȼ
+ɋ
ɋ
+ɋ
Ⱥȼ
+ɋ
Ⱥɋ
+ɋ
ȼɋ
+ɋ
Ⱥȼɋ
z
ɋ
ɯ
.
Ɂɧɚɯɨɞɠɟɧɧɹ ɞɢɫɩɟɪɫɿɣ ɫɩɿɥɶɧɨɝɨ ɜɩɥɢɜɭ ɮɚɤɬɨɪɿɜ ɋ
Ⱥȼ
, ɋ
Ⱥɋ
,
ɋ
ȼɋ
, ɋ
Ⱥȼɋ
ɜɢɦɚɝɚɽ ɨɫɨɛɥɢɜɢɯ ɩɪɢɣɨɦɿɜ, ɹɤɿ ɛɭɞɭɬɶ ɪɨɡɝɥɹɧɭɬɿ ɩɿɡɧɿɲɟ
ɧɚ ɦɚɬɟɪɿɚɥɚɯ ɤɨɧɤɪɟɬɧɢɯ ɞɚɧɢɯ.
ȼɢɜɱɚɸɱɢ ɦɟɬɨɞɢɱɧɭ ɫɬɨɪɨɧɭ ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɚɧɚɥɿɡɭ, ɦɨɠɧɚ
ɜɢɞɿɥɢɬɢ ɱɨɬɢɪɢ ɟɬɚɩɢ ɣɨɝɨ ɡɞɿɣɫɧɟɧɧɹ : 1) ɨɛɪɨɛɤɚ ɫɬɚɬɢɫɬɢɱɧɨɝɨ
ɤɨɦɩɥɟɤɫɭ ɞɥɹ ɨɞɟɪɠɚɧɧɹ ɡɚɝɚɥɶɧɨʀ ɮɚɤɬɨɪɧɨʀ ɿ ɡɚɥɢɲɤɨɜɨʀ ɞɢɫɩɟɪɫɿɣ
(ɋ
ɭ
, ɋ
ɯ
, ɋ
z
) ; 2 ) ɜɢɡɧɚɱɟɧɧɹ ɱɚɫɬɤɢ ɤɨɠɧɨʀ ɨɤɪɟɦɨʀ ɞɢɫɩɟɪɫɿʀ ɜ
ɡɚɝɚɥɶɧɿɣ, ɞɥɹ ɱɨɝɨ ɪɨɡɪɚɯɨɜɭɸɬɶ ɜɟɥɢɱɢɧɢ
2
ɯ
K
,
2
z
K
; 3) ɤɨɪɢɝɭɜɚɧɧɹ
ɨɞɟɪɠɚɧɢɯ ɞɢɫɩɟɪɫɿɣ ɧɚ ɱɢɫɥɨ ɫɬɭɩɟɧɿɜ ɜɿɥɶɧɨɫɬɿ, ɹɤɿ ɡɧɚɯɨɞɹɬɶ ɞɥɹ
ɤɨɠɧɨʀ ɞɢɫɩɟɪɫɿʀ ɡɚ ɩɟɜɧɢɦɢ ɮɨɪɦɭɥɚɦɢ; 4) ɨɰɿɧɤɚ ɮɚɤɬɨɪɧɨʀ
ɞɢɫɩɟɪɫɿʀ, ɬɨɛɬɨ ɜɫɬɚɧɨɜɥɟɧɧɹ ɜɿɪɨɝɿɞɧɨɫɬɿ ɜɩɥɢɜɭ ɤɨɠɧɨɝɨ ɜɢɛɪɚɧɨɝɨ
ɮɚɤɬɨɪɚ (ɯ ɚɛɨ Ⱥ, ȼ, ɋ ɿ ɬ.ɞ. ) ɧɚ ɜɚɪɸɸɱɭ ɨɡɧɚɤɭ. Ⱦɥɹ ɰɶɨɝɨ
ɜɢɤɨɪɢɫɬɨɜɭɸɬɶ ɤɪɢɬɟɪɿʀ Ɏɿɲɟɪɚ (F – ɤɪɢɬɟɪɿɣ).
əɤ ɡɚɡɧɚɱɚɥɨɫɹ ɜɢɳɟ, ɜ ɧɟɨɪɬɨɝɨɧɚɥɶɧɢɯ ɤɨɦɩɥɟɤɫɚɯ ɫɢɫɬɟɦɚ
ɪɨɡɪɚɯɭɧɤɭ ɱɚɫɬɤɨɜɢɯ ɞɢɫɩɟɪɫɿɣ ɋ
Ⱥ
, ɋ
ȼ
, ɋ
Ⱥȼ
ɿ ɬ.ɞ. ɦɚɽ ɞɟɹɤɿ
ɨɫɨɛɥɢɜɨɫɬɿ, ɡɭɦɨɜɥɟɧɿ ɧɟɩɪɨɩɨɪɰɿɣɧɢɦɢ ɫɩɿɜɜɿɞɧɨɲɟɧɧɹɦɢ ɱɚɫɬɨɬ
ɨɞɧɨɝɨ ɮɚɤɬɨɪɚ ɩɨ ɝɪɭɩɚɯ (ɝɪɚɞɚɰɿɹɯ) ɞɪɭɝɨɝɨ ɮɚɤɬɨɪɚ . Ⱦɥɹ ɭɫɭɧɟɧɧɹ
ɧɟɨɪɬɨɝɨɧɚɥɶɧɨɫɬɿ ɩɟɪɟɬɜɨɪɸɸɬɶ ɧɟɪɿɜɧɨɦɿɪɧɢɣ ɤɨɦɩɥɟɤɫ ɭ
ɪɿɜɧɨɦɿɪɧɢɣ . ɿɫɧɭɸɬɶ ɪɿɡɧɿ ɫɩɨɫɨɛɢ ɩɟɪɟɬɜɨɪɟɧɧɹ ɧɟɪɿɜɧɨɦɿɪɧɨɝɨ
ɤɨɦɩɥɟɤɫɭ ɜ ɪɿɜɧɨɦɿɪɧɢɣ – ɩɨ ɉɨɦɨɪɫɶɤɨɦɭ ɘ.ɉ., ɇɟɦɱɢɧɨɜɭ ȼ.ɋ. ɬɚ
ɿɧ. Ɍɨɱɧɿɫɬɶ ɰɢɯ ɦɟɬɨɞɿɜ ɞɟɹɤɨɸ ɦɿɪɨɸ ɪɿɡɧɚ, ɚɥɟ ɦɨɠɧɚ
ɤɨɪɢɫɬɭɜɚɬɢɫɹ ɛɭɞɶ – ɹɤɢɦ ɡ ɧɢɯ.
ɋɟɪɟɞ ɞɿɸɱɢɯ ɫɩɨɫɨɛɿɜ ɩɟɪɟɬɜɨɪɟɧɧɹ ɜ ɩɪɚɤɬɢɰɿ ɟɤɨɧɨɦɿɱɧɨɝɨ
ɚɧɚɥɿɡɭ ɧɚɣɛɿɥɶɲ ɡɪɭɱɧɢɦ ɫɥɿɞ ɜɢɡɧɚɬɢ ɦɟɬɨɞ ɉɨɦɨɪɫɶɤɨɝɨ ɘ.ɉ. ȼ
ɨɫɧɨɜɿ ɰɶɨɝɨ ɩɪɢɣɨɦɭ ɥɟɠɢɬɶ ɭɫɟɪɟɞɧɟɧɧɹ ɱɚɫɬɨɬ ɩɨ ɝɪɭɩɚɯ ɮɚɤɬɨɪɿɜ.
ɉɨ ɤɨɠɧɿɣ ɝɪɭɩɿ ɞɨɫɥɿɞɠɭɜɚɧɨɝɨ ɮɚɤɬɨɪɚ ɡɧɚɯɨɞɹɬɶ ɱɚɫɬɤɨɜɿ ɫɟɪɟɞɧɿ ɡ
ɜɚɪɿɚɧɬ ɩɿɞɝɪɭɩ ɿ ɡɚɝɚɥɶɧɭ ɫɟɪɟɞɧɸ ɩɨ ɜɫɶɨɦɭ ɞɢɫɩɟɪɫɿɣɧɨɦɭ
ɤɨɦɩɥɟɤɫɭ. ɋɩɨɱɚɬɤɭ ɱɚɫɬɤɨɜɿ ɿ ɡɚɝɚɥɶɧɭ ɫɟɪɟɞɧɸ ɩɿɞɧɨɫɹɬɶ ɞɨ
ɤɜɚɞɪɚɬɭ, ɚ ɩɨɬɿɦ ɰɿ ɜɟɥɢɱɢɧɢ ɩɿɞɫɬɚɜɥɹɸ
ɬɶ ɭ ɮɨɪɦɭɥɢ ɞɢɫɩɟɪɫɿɣ ɋ
ɯ
,
ɋ
Ⱥ
, ɋ
ȼ
, ɋ
ɋ
, ɋ
Ⱥȼ
ɿ ɬ.ɞ.
ɉɪɢ ɨɛɪɨɛɰɿ ɬɪɢɮɚɤɬɨɪɧɨɝɨ ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɧɟɪɿɜɧɨɦɿɪɧɨɝɨ
ɤɨɦɩɥɟɤɫɭ ɮɨɪɦɭɥɚ ɞɢɫɩɟɪɫɿʀ ɮɚɤɬɨɪɿɜ ɦɚɽ ɜɢɝɥɹɞ:
¸
¸
¹
·
¨
¨
©
§
¦
2
2
ɡɚɝ
CBA
x
ɯ
M
lll
M
nɋ
,
ɞɟ ɩ – ɱɢɫɥɨ ɨɞɢɧɢɰɶ ɫɩɨɫɬɟɪɟɠɟɧɧɹ, ɜɜɟɞɟɧɢɯ ɭ ɤɨɦɩɥɟɤɫ;
¦
2
ɯ
Ɇ
- ɫɭɦɚ ɤɜɚɞɪɚɬɿɜ ɱɚɫɬɤɨɜɢɯ ɫɟɪɟɞɧɿɯ ɚɪɢɮɦɟɬɢɱɧɢɯ ɩɨ ɜɫɿɯ
161
162
ɝɪɭɩɚɯ ɿ ɩɨ ɜɫɿɯ ɮɚɤɬɨɪɚɯ ;
CBA
lll ,,
- ɱɢɫɥɨ ɝɪɭɩ ɩɨ ɤɨɠɧɨɦɭ ɡ ɮɚɤɬɨɪɿɜ;
2
ɡɚɝ
Ɇ
- ɤɜɚɞɪɚɬ ɫɟɪɟɞɧɶɨʀ ɚɪɢɮɦɟɬɢɱɧɨʀ, ɳɨ ɪɨɡɪɚɯɨɜɭɽɬɶɫɹ ɞɥɹ ɜɫɶɨɝɨ
ɤɨɦɩɥɟɤɫɭ.
ɑɚɫɬɤɨɜɿ ɞɢɫɩɟɪɫɿʀ ɋ
Ⱥ
, ɋ
ȼ
, ɋ
ɋ
ɨɛɱɢɫɥɸɸɬɶ ɡɚ ɬɚɤɨɸ ɮɨɪɦɭɥɨɸ:
,
2
.
2
¸
¸
¹
·
¨
¨
©
§
ɡɚɝ
Ⱥ
Ⱥ
Ⱥ
Ɇ
l
Ɇ
ɩɋ
ɞɟ
¦
2
ɯ
Ɇ
-ɫɭɦɚ ɤɜɚɞɪɚɬɿɜ ɫɟɪɟɞɧɿɯ ɚɪɢɮɦɟɬɢɱɧɢɯ ɩɨ ɝɪɭɩɚɯ
ɮɚɤɬɨɪɿɜ Ⱥ, ɚ ɜɿɞɧɨɲɟɧɧɹ
Ⱥ
Ⱥ
l
Ɇ
¦
2
ɞɨɪɿɜɧɸɽ
Ⱥ
h
.Ⱥɧɚɥɨɝɿɱɧɨ ɪɨɡɪɚɯɨɜɭɸɬɶ
ɱɚɫɬɤɨɜɿ ɞɢɫɩɟɪɫɿʀ ɋ
ȼ
, ɿ ɋ
ɋ
. ȼɢɡɧɚɱɢɜɲɢ ɮɚɤɬɨɪɧɭ (ɋ
ɯ
,) ɿ ɱɚɫɬɤɨɜɿ
(ɋ
Ⱥ
, ɋ
ȼ
, ɋ
ɋ
) ɞɢɫɩɟɪɫɿʀ, ɦɨɠɧɚ ɪɨɡɪɚɯɭɜɚɬɢ ɞɢɫɩɟɪɫɿʀ ɪɿɡɧɢɯ
ɫɩɨɥɭɱɟɧɶ ɮɚɤɬɨɪɿɜ (
ȺȼɋȼɋȺɋȺȼ
ɋɋɋɋ ,,,
).
ȼɟɥɢɱɢɧɚ ɡɚɝɚɥɶɧɨʀ ɞɢɫɩɟɪɫɿʀ ɞɨɪɿɜɧɸɜɚɬɢɦɟ
=
ɋɋɋ
ɯɭ
.
ȼɟɥɢɱɢɧɚ ɡɚɝɚɥɶɧɨʀ ɞɢɫɩɟɪɫɿʀ ɿɡ ɞɚɧɨɝɨ ɜɢɪɚɡɭ ɛɭɞɟ ɞɟɳɨ ɜɿɞɪɿɡɧɹɬɢɫɹ
ɜɿɞ ɬɿɽʀ ɜɟɥɢɱɢɧɢ, ɹɤɚ ɨɞɟɪɠɚɧɚ ɡɚ ɜɢɳɟ ɧɚɜɟɞɟɧɢɦɢ ɪɨɡɪɚɯɭɧɤɚɦɢ,
ɚɥɟ ɰɹ ɪɨɡɛɿɠɧɿɫɬɶ ɞɿɽ ɧɟɫɭɬɬɽɜɨ ɧɚ ɧɚɫɬɭɩɧɿ ɨɛɱɢɫɥɟɧɧɹ. Ɉɬɠɟ,
ɧɚɜɟɞɟɧɢɦ ɫɩɨɫɨɛɨɦ ɪɨɡɪɚɯɭɧɤɿɜ ɰɿɥɤɨɦ ɦɨɠɧɚ ɤɨɪɢɫɬɭɜɚɬɢɫɹ.
ɉɪɢɧɰɢɩ ɪɨɡɪɚɯɭɧɤɭ ɨɞɧɨɮɚɤɬɨɪɧɨɝɨ ɿ ɛɚɝɚɬɨɮɚɤɬɨɪɧɨɝɨ
ɞɢɫɩɟɪɫɿɣɧɢɯ ɤɨɦɩɥɟɤɫɿɜ ɦɚɽ ɞɟɹɤɿ ɦɟɬɨɞɢɱɧɿ ɨɫɨɛɥɢɜɨɫɬɿ, ɡɭɦɨɜɥɟɧɿ
ɤɿɥɶɤɿɫɬɸ ɮɚɤɬɨɪɿɜ, ɜɤɥɸɱɟɧɢɯ ɜ ɚɧɚɥɿɡ. ɐɿ ɨɫɨɛɥɢɜɨɫɬɿ ɛɭɞɭɬɶ
ɜɢɤɥɚɞɟɧɿ ɭ ɥɨɝɿɱɧɿɣ ɩɨɫɥɿɞɨɜɧɨɫɬɿ ɪɨɡɝɥɹɞɭɜɚɧɢɯ ɧɢɠɱɟ ɪɨɡɪɚɯɭɧɤɿɜ
ɨɞɧɨ -, ɞɜɨ – ɿ ɬɪɢɮɚɤɬɨɪɧɢɯ ɞɢɫɩɟɪɫɿɣɧɢɯ ɤɨɦɩɥɟɤɫɿɜ.
7.1.2. Ⱥɥɝɨɪɢɬɦɢ ɪɿɲɟɧɧɹ ɞɢɫɩɟɪɫɿɣɧɢɯ ɦɨɞɟɥɟɣ
ɉɪɢɤɥɚɞ. Ɋɨɡɝɥɹɧɟɦɨ ɩɨɫɥɿɞɨɜɧɿɫɬɶ ɪɨɡɪɚɯɭɧɤɭ ɨɞɧɨɮɚɤɬɨɪɧɨɝɨ
ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɤɨɦɩɥɟɤɫɭ ɧɚ ɩɪɢɤɥɚɞɿ ɡɚɥɟɠɧɨɫɬɿ ɫɟɪɟɞɧɶɨɪɿɱɧɨɝɨ ɧɚɞɨɸ ɤɨɪɿɜ
( V) ɜɿɞ ɪɿɜɧɹ ɝɨɞɿɜɥɿ (Ⱥ) ɜ 30 (n) ɩɿɞɩɪɢɽɦɫɬɜɚɯ.
ɇɚ ɩɟɪɲɨɦɭ ɟɬɚɩɿ ɡɞɿɣɫɧɸɽɬɶɫɹ ɝɪɭɩɭɜɚɧɧɹ ɩɿɞɩɪɢɽɦɫɬɜ ɡɚ ɮɚɤɬɨɪɧɨɸ
ɨɡɧɚɤɨɸ. ɍ ɞɚɧɨɦɭ ɩɪɢɤɥɚɞɿ ɫɭɤɭɩɧɿɫɬɶ ɩɿɞɩɪɢɽɦɫɬɜɚɯ. ɩɨɞɿɥɟɧɚ ɧɚ ɬɪɢ ɝɪɭɩɢ
ɡɚ ɪɿɜɧɟɦ ɜɢɤɨɪɢɫɬɚɧɧɹ ɤɨɪɦɿɜ ɧɚ ɤɨɪɨɜɭ ɜ ɪɿɤ (Ⱥ). Ɉɛɪɨɛɤɚ ɜɢɯɿɞɧɨʀ ɿɧɮɨɪɦɚɰɿʀ
ɡɞɿɣɫɧɸɽɬɶɫɹ ɡɚ ɫɯɟɦɨɸ ɬɚɛɥɢɰɿ 47.
ɇɚ ɩɿɞɫɬɚɜɿ ɞɚɧɢɯ ɬɚɛɥɢɰɿ 47 ɡɧɚɯɨɞɢɦɨ ɡɚɝɚɥɶɧɭ (
ɭ
ɋ
), ɮɚɤɬɨɪɧɭ (
ɯ
ɋ
),
ɡɚɥɢɲɤɨɜɭ (
=
ɋ
) ɞɢɫɩɟɪɫɿʀ:
;6023913839740
)(
2
2
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n
v
Vɋ
ɭ
;2953913839433
)(
2
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¦
n
V
hɋ
ɯ
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=
hVɋ
162
163
ɋɩɿɜɜɿɞɧɨɲɟɧɧɹ ɫɤɥɚɞɨɜɢɯ ɞɢɫɩɟɪɫɿɣ (
=
ɋɋ
ɯ
,
) ɞɨ ɡɚɝɚɥɶɧɨʀ (
ɭ
ɋ
) ɩɨɤɚɡɭɽ
ɫɬɭɩɿɧɶ ɭɱɚɫɬɿ ɮɚɤɬɨɪɧɢɯ ɨɡɧɚɤ ɭ ɮɨɪɦɭɜɚɧɧɿ ɡɚɝɚɥɶɧɨʀ ʀɡɦɿɧɸɜɚɧɨɫɬɿ
ɪɟɡɭɥɶɬɚɬɢɜɧɨʀ ɨɡɧɚɤɢ. Ɍɚɤ, ɫɬɭɩɿɧɶ ɜɩɥɢɜɭ ɪɿɜɧɹ ɝɨɞɿɜɥɿ ɤɨɪɿɜ ɧɚ ʀɯ
ɩɪɨɞɭɤɬɢɜɧɿɫɬɶ ɫɬɚɧɨɜɢɬɢɦɟ:
49,0
602
295
2
ɭ
ɯ
ɯ
ɋ
ɋ
K
(49%).
ɋɬɭɩɿɧɶ ɜɩɥɢɜɭ ɫɭɦɢ ɿɧɲɢɯ ɧɟɜɪɚɯɨɜɚɧɢɯ ɮɚɤɬɨɪɿɜ ɧɚ ɪɟɡɭɥɶɬɚɬɢɜɧɭ
ɨɡɧɚɤɭ ɨɛɱɢɫɥɸɽɬɶɫɹ ɡɚ ɬɚɤɢɦ ɫɩɿɜɜɿɞɧɨɲɟɧɧɹɦ:
51,0
602
307
2
=
=
ɭ
ɋ
ɋ
K
( 51%).
Ɍɚɤɢɦ ɱɢɧɨɦ, ɭ ɪɨɡɝɥɹɧɭɬɨɦɭ ɩɪɢɤɥɚɞɿ ɮɚɤɬɨɪɧɚ ɨɡɧɚɤɚ (ɪɿɜɟɧɶ ɝɨɞɿɜɥɿ)
ɜɢɡɧɚɱɚɽ 49% ɡɚɝɚɥɶɧɨʀ ɜɚɪɿɚɰɿʀ ɪɟɡɭɥɶɬɚɬɢɜɧɨʀ ɨɡɧɚɤɢ (ɧɚɞɨɸ).
Ɍɚɛɥɢɰɹ 47
ȼɢɯɿɞɧɿ ɿ ɪɨɡɪɚɯɭɧɤɨɜɿ ɞɚɧɿ ɨɞɧɨɮɚɤɬɨɪɧɨɝɨ ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɤɨɦɩɥɟɤɫɭ
ɉɨɤɚɡɧɢɤ
ȼɢɬɪɚɬɢ ɤɨɪɦɿɜ ɧɚ 1 ɤɨɪɨɜɭ, ɰ ɤ. ɨɞ.
ɋɭɦɢ (
¦
)
1
Ⱥ
-ɞɨ 47,4
2
Ⱥ
-47,5-58,0
3
Ⱥ
- ɩɨɧɚɞ 58,0
V ɧɚɞɿɣ,
ɰ
32,60 35,71 49,09
31,00 32,44 40,17
30,58 33,82 35,14
32,52 31,27 42,72
32,40 34,54 37,68
32,02 34,69 44,41
31,38 32,21 38,15
41,28
34,40
44,63
37,31
35,95
37,86
36,16
34,40
36,02
¦
V
233,50 572,72 287,36 1083,58
ɯ
ɩ
7 16 7 30
2
)(
¦
V
49952,25 32800,19 82575,76
ɯ
ɩVh :)(
2
¦
7136,04 20500,51 11796,54 39433,09
¦
2
V
7142 20667 11931 39740
Ⱦɢɫɩɟɪɫɿɹ ɹɤ ɩɨɤɚɡɧɢɤ ɪɿɡɧɨɦɚɧɿɬɧɨɫɬɿ ɡɚɥɟɠɢɬɶ ɜɿɞ ɤɿɥɶɤɨɫɬɿ ɨɞɢɧɢɰɶ
ɫɩɨɫɬɟɪɟɠɟɧɧɹ (ɩɿɞɩɪɢɽɦɫɬɜ) ɭ ɝɪɭɩɿ. Ⱦɥɹ ɜɢɡɧɚɱɟɧɧɹ ɜɩɥɢɜɭ ɮɚɤɬɨɪɿɜ ɰɹ
ɨɛɫɬɚɜɢɧɚ ɧɟ ɦɚɽ ɡɧɚɱɟɧɧɹ. ȼ ɿɧɲɢɯ ɠɟ ɜɢɩɚɞɤɚɯ, ɡɨɤɪɟɦɚ, ɩɪɢ ɜɫɬɚɧɨɜɥɟɧɧɿ
ɜɿɪɨɝɿɞɧɨɫɬɿ ɜɩɥɢɜɭ ɮɚɤɬɨɪɿɜ, ɧɟɨɛɯɿɞɧɢɣ ɩɨɤɚɡɧɢɤ, ɜɿɥɶɧɢɣ ɜɿɞ ɜɤɚɡɚɧɨʀ
ɡɚɥɟɠɧɨɫɬɿ, ɳɨ ɞɨɩɭɫɤɚɽ ɩɨɪɿɜɧɹɧɧɹ ɝɪɭɩ, ɪɿɡɧɢɯ ɡɚ ɤɿɥɶɤɿɫɬɸ ɟɥɟɦɟɧɬɿɜ, ɳɨ
163
164
ɜɯɨɞɹɬɶ ɞɨ ɧɢɯ. Ɍɚɤɢɦ ɩɨɤɚɡɧɢɤɨɦ ɽ ɤɨɪɢɝɨɜɚɧɚ ɞɢɫɩɟɪɫɿɹ – ɞɟɜɿɚɬɚ.
Ⱦɟɜɿɚɬɨɸ ɧɚɡɢɜɚɸɬɶ ɞɢɫɩɟɪɫɿɸ, ɹɤɚ ɩɪɢɩɚɞɚɽ ɧɚ ɨɞɢɧ ɟɥɟɦɟɧɬ ɜɿɥɶɧɨɝɨ
ɜɚɪɿɸɜɚɧɧɹ ɚɛɨ ɧɚ ɨɞɢɧ ɫɬɭɩɿɧɶ ɜɿɥɶɧɨɫɬɿ.
Ʉɨɪɿɧɶ ɤɜɚɞɪɚɬɧɢɣ ɡ ɞɟɜɿɚɬɢ (
2
V
) ɹɜɥɹɽ ɫɨɛɨɸ ɡɜɢɱɚɣɧɢɣ ɩɨɤɚɡɧɢɤ
ɦɚɬɟɦɚɬɢɱɧɨʀ ɫɬɚɬɢɫɬɢɤɢ – ɫɟɪɟɞɧɽ ɤɜɚɞɪɚɬɢɱɧɟ ɜɿɞɯɢɥɟɧɧɹ (
V
).
ɍ ɧɚɲɨɦɭ ɩɪɢɤɥɚɞɿ ɱɢɫɥɨ ɫɬɭɩɟɧɿɜ ɜɿɥɶɧɨɫɬɿ ɜɚɪɿɚɰɿʀ (
Q
) ɞɥɹ ɮɚɤɬɨɪɧɨʀ
ɨɡɧɚɤɢ ɿ ɞɥɹ ɧɟɜɪɚɯɨɜɚɧɢɯ ɮɚɤɬɨɪɿɜ ɫɬɚɧɨɜɢɬɢɦɟ ɜɿɞɩɨɜɿɞɧɨ :
;2131 l
ɯ
Q
27330
=
lɩ
Q
, ɞɟ l – ɤɿɥɶɤɿɫɬɶ ɜɢɞɿɥɟɧɢɯ ɝɪɭɩ; n – ɱɢɫɟɥɶɧɿɫɬɶ ɜɢɛɿɪɤɢ.
Ɋɨɡɪɚɯɭɽɦɨ ɞɟɜɿɚɬɢ :
;50,147
2
295
2
ɯ
ɯ
ɯ
ɋ
Q
V
37,11
27
307
2
=
=
=
Q
V
ɋ
Ʉɪɢɬɟɪɿɽɦ ɜɿɪɨɝɿɞɧɨɫɬɿ ɜɩɥɢɜɭ ɮɚɤɬɨɪɧɨʀ ɨɡɧɚɤɢ ɧɚ ɪɟɡɭɥɶɬɚɬɢɜɧɭ ɽ
ɫɩɿɜɜɿɞɧɨɲɟɧɧɹ ʀʀ ɞɟɜɿɚɬɢ ɞɨ ɞɟɜɿɚɬɢ ɧɟɜɪɚɯɨɜɚɧɢɯ ɮɚɤɬɨɪɿɜ. əɤɳɨ ɪɨɡɪɚɯɨɜɚɧɟ
ɫɩɿɜɜɿɞɧɨɲɟɧɧɹ ɞɨɪɿɜɧɸɽ ɱɢ ɛɿɥɶɲɟ ɜɢɡɧɚɱɟɧɨʀ ɫɬɚɧɞɚɪɬɧɨʀ ɜɟɥɢɱɢɧɢ, ɜɩɥɢɜ
ɜɜɚɠɚɽɬɶɫɹ ɜɿɪɨɝɿɞɧɢɦ ɡ ɩɟɜɧɢɦ ɫɬɭɩɟɧɟɦ ɿɦɨɜɿɪɧɨɫɬɿ. ɋɬɚɧɞɚɪɬɧɿ ɜɿɞɧɨɲɟɧɧɹ
ɞɟɜɿɚɬ ɜɢɡɧɚɱɚɸɬɶɫɹ ɡɚ ɫɩɟɰɿɚɥɶɧɢɦɢ ɬɚɛɥɢɰɹɦɢ (ɞɨɞɚɬɤɢ 8,9).
Ɂɧɚɯɨɞɢɦɨ ɰɟ ɫɩɿɜɜɿɞɧɨɲɟɧɧɹ ɞɥɹ ɮɚɤɬɨɪɧɨʀ ɨɡɧɚɤɢ ɧɚ ɬɚɤɨɦɭ ɩɪɢɤɥɚɞɿ:
.97,12
37,11
50,147
2
2
=
V
V
ɯ
ɪ
F
Ɉɞɟɪɠɚɧɢɣ ɤɪɢɬɟɪɿɣ (
ɪ
F
) ɩɨɪɿɜɧɸɽɦɨ ɡ ɬɚɛɥɢɱɧɢɦ ɣɨɝɨ ɡɧɚɱɟɧɧɹɦ ɩɪɢ
ɞɜɨɯ ɩɨɪɨɝɚɯ ɣɦɨɜɿɪɧɨɫɬɿ : 0,95; 0,99 (ɞɨɞɚɬɤɢ 8,9).
ɇɚɜɟɞɟɦɨ ɫɬɚɧɞɚɪɬɧɿ ɫɩɿɜɜɿɞɧɨɲɟɧɧɹ ɞɟɜɿɚɬ, ɳɨ ɜɿɞɩɨɜɿɞɚɸɬɶ ɫɬɭɩɟɧɹɦ
ɜɿɥɶɧɨɫɬɿ ɜɚɪɿɚɰɿʀ.
ȱɦɨɜɿɪɧɿɫɬɶ Ɋ Ʉɪɢɬɟɪɿɣ F
0,95 3,3
0,99 5,5
ɍ ɧɚɲɨɦɭ ɩɪɢɤɥɚɞɿ
ɪ
F
(12,97) >
Ɍ
F
(3,3). Ɉɬɠɟ, ɜ ɞɨɫɥɿɞɠɭɜɚɧɢɯ
ɩɿɞɩɪɢɽɦɫɬɜɚɯ ɜɩɥɢɜ ɪɿɜɧɹ ɝɨɞɿɜɥɿ ɤɨɪɿɜ ɧɚ ʀɯ ɩɪɨɞɭɤɬɢɜɧɿɫɬɶ ɜɢɹɜɢɜɫɹ ɞɨɫɢɬɶ
ɫɢɥɶɧɢɦ ɿ ɜɿɪɨɝɿɞɧɢɦ. ɉɪɨ ɜɿɪɨɝɿɞɧɿɫɬɶ ɪɟɡɭɥɶɬɚɬɿɜ ɚɧɚɥɿɡɭ ɫɜɿɞɱɢɬɶ ɜɢɫɨɤɢɣ
ɫɬɭɩɿɧɶ ɿɦɨɜɿɪɧɨɫɬɿ 0,99.
ɉɪɢɤɥɚɞ. Ɋɨɡɪɚɯɭɧɨɤ ɞɜɨɮɚɤɬɨɪɧɨɝɨ ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɤɨɦɩɥɟɤɫɭ
ɪɨɡɝɥɹɧɟɦɨ ɧɚ ɩɪɢɤɥɚɞɿ ɜɢɜɱɟɧɧɹ ɡɚɥɟɠɧɨɫɬɿ ɫɨɛɿɜɚɪɬɨɫɬɿ 1ɰ ɦɨɥɨɤɚ (V) ɜ 30
ɩɿɞɩɪɢɽɦɫɬɜ ɪɚɣɨɧɭ ɜɿɞ ɪɿɜɧɹ ɤɨɧɰɟɧɬɪɚɰɿʀ ɩɨɝɨɥɿɜ’ɹ ɤɨɪɿɜ (Ⱥ) ɿ ɫɩɟɰɿɚɥɿɡɚɰɿʀ
ɜɢɪɨɛɧɢɰɬɜɚ ɦɨɥɨɤɚ (ȼ). ɡ ɰɿɽɸ ɦɟɬɨɸ ɫɭɤɭɩɧɿɫɬɶ ɪɨɡɞɿɥɟɧɚ ɧɚ 2 ɝɪɭɩɢ ɡ
ɩɨɞɚɥɶɲɢɦ ɩɨɞɿɥɨɦ ɧɚ 2 ɩɿɞɝɪɭɩɢ (ɬɚɛɥ. 48).
ɇɚ ɩɿɞɫɬɚɜɿ ɪɨɡɪɚɯɭɧɤɨɜɢɯ ɞɚɧɢɯ ɬɚɛɥɢɰɿ 48 ɜɢɡɧɚɱɚɽɦɨ ɡɚɝɚɥɶɧɭ (
ɭ
ɋ
),
ɮɚɤɬɨɪɧɭ (
ɯ
ɋ
) ɿ ɡɚɥɢɲɤɨɜɭ (
=
ɋ
) ɞɢɫɩɟɪɫɿʀ:
;14,532
30
)98,826(
67,23328
)(
2
2
2
¦
¦
ɯ
ɭ
n
V
Vɋ
164
165
;80,226
)98,826(
33,23023
)(
2
2
¦
¦
ɯx
ɯ
ɩn
V
hɋ
.34,30533,2302367,23328
2
¦¦
=
hVɋ
Ɍɚɛɥɢɰɹ 48
ȼɢɯɿɞɧɿ ɿ ɪɨɡɪɚɯɭɧɤɨɜɿ ɞɚɧɿ ɞɜɨɮɚɤɬɨɪɧɨɝɨ ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɤɨɦɩɥɟɤɫɭ
ɉɨɤɚɡɧɢɤ
Ʉɨɪɿɜ ɧɚ 100 ɝɚ ɫ.-ɝ. ɭɝɿɞɶ, ɝɨɥ.
ɋɭɦɚ
(
¦
)
1
Ⱥ
- ɞɨ 50
2
Ⱥ
- ɩɨɧɚɞ 50
ɉɢɬɨɦɚ ɜɚɝɚ ɦɨɥɨɤɚ ɜ ɨɛɫɹɡɿ ɬɨɜɚɪɧɨʀ
ɩɪɨɞɭɤɰɿʀ, %
1
ȼ
-ɞɨ 30
2
ȼ
-ɩɨɧɚɞ
30
1
ȼ
-ɞɨ 30
2
ȼ
-ɩɨɧɚɞ
30
V
(ɋɨɛɿɜɚɪɬɿɫɬɶ 1ɰ
ɦɨɥɨɤɚ, ɝɪɧ.)
27,59 35,65 19,45 23,49
32,89 34,20 17,90 28,86
27,23 29,50 19,84 25,46
35,30 28,19 22,26 26,96
25,02 27,84 25,83 28,82
29,44 28,75 27,20 28,27
26,19 31,57 25,47 31,87
26,87 29,07
¦
V
230,53 215,70 187,02 193,73 826,98
ɯ
ɩ
8 7 8 7 30
2
)(
¦
V
53144,08 46526,49 34976,48 37531,31 -
ɯ
ɩVh :)(
2
¦
6643,01 6646,64 4372,06 5361,62 23023,33
¦
ɯɯ
ɩVɆ :
28,81 30,81 23,38 27,68 -
2
ɯ
Ɇ
830,01 949,25 546,62 766,18 3092,06
2
¦
V
… … … … 23328,67
ɋɬɭɩɿɧɶ ɜɩɥɢɜɭ ɮɚɤɬɨɪɧɢɯ ɨɡɧɚɤ (ɤɨɧɰɟɧɬɪɚɰɿʀ ʀʀ ɿ ɫɩɟɰɿɚɥɿɡɚɰɿʀ
ɜɢɪɨɛɧɢɰɬɜɚ) ɧɚ ɪɟɡɭɥɶɬɚɬɢɜɧɭ ɨɡɧɚɤɭ (ɫɨɛɿɜɚɪɬɿɫɬɶ ɜɢɪɨɛɧɢɰɬɜɚ ɦɨɥɨɤɚ)
ɫɬɚɧɨɜɢɬɢɦɟ :
4262,0
14,532
80,226
2
ɭ
ɯ
ɯ
ɋ
ɋ
K
(42,6 %). ɋɬɭɩɿɧɶ ɜɩɥɢɜɭ ɧɟɜɪɚɯɨɜɚɧɢɯ
ɮɚɤɬɨɪɿɜ ɧɚ ɪɟɡɭɥɶɬɚɬɢɜɧɭ ɨɡɧɚɤɭ ɛɭɞɟ:
5738,0
14,532
34,305
=
=
ɭ
ɋ
ɋ
K
(57,4 %).
Ⱦɥɹ ɤɿɥɶɤɿɫɧɨʀ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɜɩɥɢɜɭ ɤɨɠɧɨɝɨ ɡ ɮɚɤɬɨɪɿɜ ɫɥɿɞ ɞɢɫɩɟɪɫɿɸ
ʀɯ ɫɭɦɚɪɧɨʀ ɞɿʀ ɪɨɡɤɥɚɫɬɢ ɧɚ ɫɤɥɚɞɨɜɿ, ɬɨɛɬɨ ɜ ɮɚɤɬɨɪɧɿɣ ɞɢɫɩɟɪɫɿʀ (
ɯ
ɋ
)
ɜɢɞɿɥɢɬɢ ɞɢɫɩɟɪɫɿʀ ɩɟɪɲɨɝɨ (
Ⱥ
ɋ
) ɿ ɞɪɭɝɨɝɨ (
ȼ
ɋ
) ɮɚɤɬɨɪɿɜ, ɚ ɬɚɤɨɠ ʀɯ ɫɩɨɥɭɱɟɧɧɹ
(
Ⱥȼ
ɋ
).Ⱦɢɫɩɟɪɫɿɹ
Ⱥȼ
ɋ
ɯɚɪɚɤɬɟɪɢɡɭɽ ɫɬɭɩɿɧɶ ɡɭɦɨɜɥɟɧɨɫɬɿ ɜɩɥɢɜɭ ɩɟɪɲɨɝɨ
ɮɚɤɬɨɪɚ ɞɪɭɝɢɦ. ɋɤɥɚɞɟɦɨ ɞɨɩɨɦɿɠɧɭ ɬɚɛɥɢɰɸ 49. Ɍɪɟɬɹ ɤɨɥɨɧɤɚ ɰɿɽʀ ɬɚɛɥɢɰɿ
ɪɨɡɪɚɯɨɜɭɽɬɶɫɹ ɧɚ ɨɫɧɨɜɿ ɞɚɧɢɯ ɬɚɛɥɢɰɿ 49.
59,62=28,81 +30,81; 52,19 = 28,81 + 23,38
165
166
Ɍɚɛɥɢɰɹ 49
Ⱦɨɩɨɦɿɠɧɿ ɪɨɡɪɚɯɭɧɤɢ ɞɥɹ ɨɛɪɨɛɤɢ ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɤɨɦɩɥɟɤɫɭ
ɡɚ ɮɚɤɬɨɪɚɦɢ Ⱥ, ȼ
Ƚɪɚɞɚɰɿɹ, ɿ ɑɢɫɥɨ ɫɟɪɟɞɧɿɯ , l
¦
0
ɯ
l
ɯ
ɿ
¦
0
0
2
l
0
1
Ⱥ
2 59,69 29,81 888,63
2
Ⱥ
2 51,06 25,53 651,78
110,68
41,1540
2
0
¦
Ⱥ
1
ȼ
2 52,19 26,10 681,2100
2
ȼ
2 58,49 29,24 854,9776
110,68
1876,1536
2
0
¦
ȼ
ɋɟɪɟɞɧɹ ɚɪɢɮɦɟɬɢɱɧɚ (ɡɚɝɚɥɶɧɚ) ɩɨ ɝɪɚɞɚɰɿɹɯ ɮɚɤɬɨɪɿɜ ɫɬɚɧɨɜɢɬɶ:
.67,27
4
68,110
0
0
¦
l
ɯ
ɡɚɝ
ɋɬɭɩɿɧɶ ɪɿɡɧɨɦɚɧɿɬɧɨɫɬɿ ɫɟɪɟɞɧɿɯ ɚɪɢɮɦɟɬɢɱɧɢɯ ɫɨɛɿɜɚɪɬɨɫɬɿ 1ɰ ɦɨɥɨɤɚ
ɪɨɡɪɚɯɨɜɭɽɦɨ ɜ ɬɚɤɿɣ ɩɨɫɥɿɞɨɜɧɨɫɬɿ :
ɞɥɹ ɜɫɿɯ ɝɪɚɞɚɰɿɣ –
;70,22167,27
22
06,3092
30
22
2
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
©
§
0
0
c
¦
ɡɚɝ
ȼA
ɯ
ɯ
ll
ɩɋ
ɞɥɹ ɝɪɚɞɚɰɿʀ
Ⱥ
ɋȺ
c
=
10,13767,27
2
41,1540
30
22
2
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
©
§
0
0
¦
ɡɚɝ
Ⱥ
Ⱥ
l
ɩ
;
ɞɥɹ ɝɪɚɞɚɰɿʀ
10,7467,27
2
19,1536
30
22
2
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
©
§
0
0
c
¦
ɡɚɝ
ȼ
ȼ
ȼ
l
ɩɋȼ
;
ɞɥɹ ɫɩɨɥɭɱɟɧɧɹ ɮɚɤɬɨɪɿɜ
Ⱥ ɿ
c
c
c
c
ȼȺɯȺȼ
ɋɋɋɋȼ
221,70 – 137,10 – 74,10 =10,50
Ⱦɥɹ ɪɨɡɤɥɚɞɭ ɫɭɦɚɪɧɨʀ ɞɢɫɩɟɪɫɿʀ ɞɨɫɥɿɞɠɭɜɚɧɢɯ ɮɚɤɬɨɪɿɜ ɧɚ ɫɤɥɚɞɨɜɿ
ɪɨɡɪɚɯɨɜɭɽɦɨ ɩɨɩɪɚɜɨɱɧɢɣ ɤɨɟɮɿɰɿɽɧɬ:
.023,1
70,221
80,226
c
ɯ
ɯ
ɋ
ɋ
Ʉ
Ⱦɢɫɩɟɪɫɿʀ, ɡɭɦɨɜɥɟɧɿ ɞɿɽɸ ɞɨɫɥɿɞɠɭɜɚɧɢɯ ɮɚɤɬɨɪɿɜ ɿ ʀɯ ɫɩɨɥɭɱɟɧɧɹ,
ɫɬɚɧɨɜɥɹɬɶ:
Ʉɋɋ
ȺȺ
c
=
;25,140023,110,137
ȼ
ɋ
=
;81,75023,110,74
c
Ʉɋ
ȼ
.74,10023,150,10
c
Ʉɋɋ
ȺȼȺȼ
Ʉɿɧɰɟɜɚ ɞɢɫɩɟɪɫɿɣɧɚ ɫɬɪɭɤɬɭɪɚ ɞɜɨɮɚɤɬɨɪɧɨɝɨ ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɤɨɦɩɥɟɤɫɭ
ɦɚɬɢɦɟ ɜɢɝɥɹɞ:
.14,532034,30580,226)(
=
ɋɋɋɋɋ
ȺȼȼȺɭ
Ɋɨɡɪɚɯɨɜɭɽɦɨ ɫɬɭɩɿɧɶ ɜɩɥɢɜɭ ɮɚɤɬɨɪɿɜ, ɳɨ ɜɢɜɱɚɸɬɶɫɹ, ɧɚ ɮɨɪɦɭɜɚɧɧɹ
ɡɦɿɧɸɜɚɧɨɫɬɿ ɪɟɡɭɥɶɬɚɬɢɜɧɨʀ ɨɡɧɚɤɢ. Ɂɨɤɪɟɦɚ, ɪɿɜɟɧɶ ɤɨɧɰɟɧɬɪɚɰɿʀ ɩɨɝɨɥɿɜ’ɹ
ɤɨɪɿɜ ɜɢɡɧɚɱɚɽ ɜɚɪɿɚɰɿɸ ɫɨɛɿɜɚɪɬɨɫɬɿ 1 ɰ ɦɨɥɨɤɚ
,2636,0
14,532
25,140
2
ɭ
Ⱥ
Ⱥ
ɋ
ɋ
K
ɚɛɨ
26,36 %; ɮɚɤɬɨɪ ɫɩɟɰɿɚɥɿɡɚɰɿʀ –
,1425,0
14,532
81,75
2
ɭ
ȼ
ȼ
ɋ
ɋ
K
ɚɛɨ 14,25; ɜɡɚɽɦɨɞɿɹ
166
167
ɮɚɤɬɨɪɿɜ –
,0202,0
14,532
74,10
2
ɭ
Ⱥȼ
Ⱥȼ
ɋ
ɋ
K
ɚɛɨ 2,02 %.
ɑɢɫɥɚ ɫɬɭɩɟɧɿɜ ɜɿɥɶɧɨɫɬɿ ɞɥɹ ɪɨɡɪɚɯɭɧɤɭ ɞɟɜɿɚɬɢ ɜ ɞɜɨɮɚɤɬɨɪɧɨɦɭ
ɤɨɦɩɥɟɤɫɿ ɪɨɡɪɚɯɨɜɭɸɬɶɫɹ ɜ ɬɚɤɿɣ ɩɨɫɥɿɞɨɜɧɨɫɬɿ :
;1121
AȺ
l
Q
;1121
ȼȼ
l
Q
;1
ȼAȺȼ
ll
Q
;3
ȺȼȼȺɯ
Q
Q
Q
Q
.26430
= ȼȺ
llɩ
Q
ȼɢɡɧɚɱɚɽɦɨ ɞɟɜɿɚɬɢ :
;25,140
1
25,140
2
Ⱥ
Ⱥ
Ⱥ
ɋ
Q
V
;81,75
1
81,75
2
ȼ
ȼ
ȼ
ɋ
Q
V
;74,10
1
74,10
2
Ⱥȼ
Ⱥȼ
Ⱥȼ
ɋ
Q
V
;60,75
3
80,226
2
ɯ
ɯ
ɯ
ɋ
Q
V
.74,11
26
34,305
2
=
=
Q
V
Ⱥȼ
ɋ
Ɋɨɡɪɚɯɨɜɭɽɦɨ F –ɤɪɢɬɟɪɿɣ:
;96,11
74,11
25,140
2
2
=
V
V
A
A
F
;45,6
74,11
81,75
2
2
=
V
V
ȼ
ȼ
F
;91,0
74,11
74,10
2
2
=
V
V
Aȼ
Aȼ
F .44,6
74,11
60,75
2
2
=
V
V
ɯ
ɯ
F
Ɉɞɟɪɠɚɧɿ ɤɪɢɬɟɪɿʀ ɩɨɪɿɜɧɸɽɦɨ ɡ ɬɚɛɥɢɱɧɢɦɢ ʀɯ ɡɧɚɱɟɧɧɹɦɢ ɩɪɢ ɞɜɨɯ
ɩɨɪɨɝɚɯ ɿɦɨɜɿɪɧɨɫɬɿ 0,95 ɿ 0,99 (ɞɨɞɚɬɤɢ 8,9).
ɋɬɚɧɞɚɪɬɧɿ ɜɿɞɧɨɲɟɧɧɹ ɞɟɜɿɚɬ, ɹɤɿ ɜɿɞɩɨɜɿɞɚɸɬɶ ɫɬɭɩɟɧɹɦ ɜɿɥɶɧɨɫɬɿ
ɜɚɪɿɚɰɿʀ ɧɟɜɪɚɯɨɜɚɧɢɯ ɮɚɤɬɨɪɿɜ (
26
=
Q
) ɿ ɪɨɡɪɚɯɨɜɚɧɢɦ ɜɢɳɟ ɫɬɭɩɟɧɹɦ
ɜɿɥɶɧɨɫɬɿ ɜɚɪɿɚɰɿʀ ɞɨɫɥɿɞɠɭɜɚɧɢɯ ɮɚɤɬɨɪɿɜ ɫɬɚɧɨɜɢɬɢɦɭɬɶ:
ȱɦɨɜɿɪɧɿɫɬɶ Ɋ Ʉɪɢɬɟɪɿɣ F
ɉɪɢ
1
1
Q
ɉɪɢ
3
3
Q
0,95 4,22 2,98
0,99 7,72 4,64
Ɋɟɡɭɥɶɬɚɬɢ ɚɧɚɥɿɡɭ ɤɨɠɧɨɝɨ ɡ ɮɚɤɬɨɪɿɜ ɨɤɪɟɦɨʀ ɱɢ ɫɭɦɚɪɧɨʀ ʀɯ ɞɿʀ ɫɥɿɞ
ɜɜɚɠɚɬɢ ɜɿɪɨɝɿɞɧɢɦɢ ɩɪɢ ɬɢɯ ɩɨɪɨɝɚɯ ɿɦɨɜɿɪɧɨɫɬɿ, ɞɟ
ɪ
F
>
Ɍ
F
. ɇɟɞɨɫɬɨɜɿɪɧɢɦɢ ɩɪɢ
ɪ
F
<
Ɍ
F
.
ɉɪɢɤɥɚɞ. Ɋɨɡɝɥɹɧɟɦɨ ɩɨɫɥɿɞɨɜɧɿɫɬɶ ɪɨɡɪɚɯɭɧɤɭ ɬɪɢɮɚɤɬɨɪɧɨɝɨ
ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɤɨɦɩɥɟɤɫɭ ɧɚ ɩɪɢɤɥɚɞɿ ɜɢɜɱɟɧɧɹ ɡɚɥɟɠɧɨɫɬɿ ɫɨɛɿɜɚɪɬɨɫɬɿ
ɜɢɪɨɛɧɢɰɬɜɚ 1ɰ ɹɥɨɜɢɱɢɧɢ (V) ɜɿɞ ɪɿɜɧɹ ɩɪɨɞɭɤɬɢɜɧɨɫɬɿ ɩɪɚɰɿ (Ⱥ) , ɪɿɜɧɹ ɜɢɬɪɚɬ
ɤɨɪɦɿɜ ɧɚ 1 ɰ ɩɪɢɪɨɫɬɭ (ȼ) ɿ ɫɨɛɿɜɚɪɬɨɫɬɿ 1ɰ ɤɨɪɦɨɜɢɯ ɨɞɢɧɢɰɶ (ɋ). Ɂ ɦɟɬɨɸ
ɤɿɥɶɤɿɫɧɨʀ ɨɰɿɧɤɢ ɧɚɡɜɚɧɢɯ ɜɢɳɟ ɮɚɤɬɨɪɿɜ ɧɚ ɪɟɡɭɥɶɬɚɬɢɜɧɭ ɨɡɧɚɤɭ ɛɭɞɭɽɦɨ
ɬɪɢɮɚɤɬɨɪɧɢɣ ɞɢɫɩɟɪɫɿɣɧɢɣ ɤɨɦɩɥɟɤɫ, ɨɫɧɨɜɭ ɹɤɨɝɨ ɫɬɚɧɨɜɢɬɶ ɤɨɦɛɿɧɚɰɿɣɧɟ
ɝɪɭɩɭɜɚɧɧɹ 66 ɩɿɞɩɪɢɽɦɫɬɜ (ɬɚɛɥ. 50).
Ⱦɨɫɥɿɞɠɭɜɚɧɭ ɫɭɤɭ
ɩɧɿɫɬɶ ɫɩɨɱɚɬɤɭ ɪɨɡɩɨɞɿɥɟɧɨ ɧɚ ɞɜɿ ɝɪɭɩɢ: ɡ ɪɿɜɧɟɦ
ɡɚɬɪɚɬ ɪɨɛɨɱɨɝɨ ɱɚɫɭ ɧɚ ɜɢɪɨɛɧɢɰɬɜɨ 1 ɰ ɹɥɨɜɢɱɢɧɢ ɞɨ 90 ɥɸɞɢɧɨ –ɝɨɞɢɧ. (
1
Ⱥ
) ɿ
ɩɨɧɚɞ 90 ɥɸɞɢɧɨ –ɝɨɞɢɧ (
2
Ⱥ
). ɍ ɤɨɠɧɿɣ ɝɪɭɩɿ ɛɭɥɨ ɜɢɞɿɥɟɧɨ ɩɨ ɞɜɿ ɩɿɞɝɪɭɩɢ ɡ
ɫɟɪɟɞɧɿɦ ɪɨɡɦɿɪɨɦ ɜɢɬɪɚɬ ɤɨɪɦɿɜ ɧɚ 1 ɰ ɩɪɨɞɭɤɰɿʀ: ɦɟɧɲɟ 10 (
1
ȼ
) ɿ ɛɿɥɶɲɟ 10 ɰ
167
168
ɤɨɪɦɨɜɢɯ ɨɞɢɧɢɰɶ. (
2
ȼ
). ɉɨɬɿɦ ɤɨɠɧɚ ɡ ɧɢɯ ɭ ɫɜɨɸ ɱɟɪɝɭ ɪɨɡɩɨɞɿɥɟɧɚ ɳɟ ɧɚ ɞɜɿ
ɩɿɞɝɪɭɩɢ: ɡ ɫɨɛɿɜɚɪɬɿɫɬɸ 1 ɰ ɤɨɪɦɨɜɢɯ ɨɞɢɧɢɰɶ, ɡɝɨɞɨɜɚɧɢɯ ɬɜɚɪɢɧɚɦ, ɞɨ 14
ɝɪɧ.(
1
ɋ
) ɿ ɩɨɧɚɞ 14 ɝɪɧ. (
2
ɋ
). ɍ ɪɟɡɭɥɶɬɚɬɿ ɞɨɫɥɿɞɠɭɜɚɧɚ ɫɭɤɭɩɧɿɫɬɶ ɩɿɞɩɪɢɽɦɫɬɜ
ɛɭɥɚ ɪɨɡɩɨɞɿɥɟɧɚ ɧɚ 8 ɩɿɞɝɪɭɩ, ɩɨ ɤɨɠɧɿɣ ɡ ɹɤɢɯ ɧɚɜɟɞɟɧɨ ɜɚɪɿɚɧɬɢ
ɪɟɡɭɥɶɬɚɬɢɜɧɨʀ ɨɡɧɚɤɢ (V) – ɪɿɜɟɧɶ ɫɨɛɿɜɚɪɬɨɫɬɿ ɜɢɪɨɛɧɢɰɬɜɚ 0,1ɰ ɹɥɨɜɢɱɢɧɢ.
Ɉɫɤɿɥɶɤɢ ɭ ɧɚɲɨɦɭ ɩɪɢɤɥɚɞɿ ɪɨɡɝɥɹɞɚɽɬɶɫɹ ɬɪɢɮɚɤɬɨɪɧɢɣ ɧɟɪɿɜɧɨɦɿɪɧɢɣ
ɤɨɦɩɥɟɤɫ, ɨɛɪɨɛɤɭ ɣɨɝɨ ɡɞɿɣɫɧɸɽɦɨ ɜ ɬɚɤɿɣ ɩɨɫɥɿɞɨɜɧɨɫɬɿ: ɫɩɨɱɚɬɤɭ ɛɭɞɭɽɦɨ
ɡɜɢɱɚɣɧɢɦ ɱɢɧɨɦ ɤɨɪɟɥɹɰɿɣɧɭ ɪɟɲɿɬɤɭ, ɩɨɬɿɦ ɜɢɤɨɧɭɽɦɨ ɞɨɩɨɦɿɠɧɿ
ɪɨɡɪɚɯɭɧɤɢ, ɪɟɡɭɥɶɬɚɬɢ ɹɤɢɯ ɡɚɧɨɫɢɦɨ ɜ ɰɸ ɠ ɬɚɛɥɢɰɸ. Ⱦɨ ɧɢɯ ɜɿɞɧɨɫɹɬɶɫɹ
ɤɿɥɶɤɿ
ɫɬɶ (ɩ=66) ɿ ɫɭɦɚ (
¦
V
) ɜɚɪɿɚɧɬ ɞɨɫɥɿɞɠɭɜɚɧɨɝɨ ɤɨɦɩɥɟɤɫɭ, ɫɭɦɚ ɱɚɫɬɨɤ
ɜɿɞ ɞɿɥɟɧɧɹ ɤɜɚɞɪɚɬɿɜ ɫɭɦ ɜɚɪɿɚɧɬ ɩɨ ɤɨɠɧɿɣ ɩɿɞɝɪɭɩɿ ɧɚ ɱɢɫɥɨ ɜɚɪɿɚɧɬ
(
¦
h
=669887,82), ɫɭɦɚ ɤɜɚɞɪɚɬɿɜ ɫɟɪɟɞɧɿɯ ɚɪɢɮɦɟɬɢɱɧɢɯ ɩɨ ɩɿɞɝɪɭɩɚɯ
¦
00.78826
2
ɯ
Ɇ
.
ɇɚ ɩɿɞɫɬɚɜɿ ɪɨɡɪɚɯɭɧɤɨɜɢɯ ɞɚɧɢɯ ɬɚɛɥɢɰɿ 50 ɜɢɡɧɚɱɚɽɦɨ ɡɚɝɚɥɶɧɭ (
ɭ
ɋ
),
ɮɚɤɬɨɪɧɭ (
ɯ
ɋ
) ɿ ɡɚɥɢɲɤɨɜɭ (
=
ɋ
) ɞɢɫɩɟɪɫɿʀ:
;20,12936
66
77,6599
20,672890
)(
2
2
2
¦
¦
n
V
V
ɋ
ɭ
;32,9933
66
77,6599
32,669887
)(
2
2
2
¦
¦
n
V
h
ɋ
ɯ
¦¦
88.300232.66988720.672890
22
hVC
z
.
ȼɫɬɚɧɨɜɥɸɽɦɨ ɱɚɫɬɤɭ ɜɫɿɯ ɞɨɫɥɿɞɠɭɜɚɧɢɯ ɮɚɤɬɨɪɿɜ ɭ ɡɚɝɚɥɶɧɿɣ ɜɚɪɿɚɰɿʀ
ɪɟɡɭɥɶɬɚɬɢɜɧɨʀ ɨɡɧɚɤɢ. Ɍɚɤ, ɫɬɭɩɿɧɶ ɜɩɥɢɜɭ ɩɪɨɞɭɤɬɢɜɧɨɫɬɿ ɩɪɚɰɿ, ɪɨɡɦɿɪɭ
ɜɢɬɪɚɬ ɿ ɜɚɪɬɨɫɬɿ ɤɨɪɦɿɜ ɧɚ ɪɿɜɟɧɶ ɫɨɛɿɜɚɪɬɨɫɬɿ ɫɬɚɧɨɜɢɬɶ :
,768,0
20,12936
32,9933
2
ɯ
K
ɚɛɨ 76,8%; ɚ ɫɭɦɢ ɧɟɜɪɚɯɨɜɚɧɢɯ ɮɚɤɬɨɪɿɜ –
,232,0
20,12936
88,3002
2
=
K
ɚɛɨ 23,2%.
əɤ ɜɿɞɡɧɚɱɚɥɨɫɹ ɜɢɳɟ, ɭ ɛɚɝɚɬɨɮɚɤɬɨɪɧɢɯ ɤɨɦɩɥɟɤɫɚɯ ɞɢɫɩɟɪɫɿɹ ɫɩɿɥɶɧɨʀ
ɞɿʀ ɜɪɚɯɨɜɚɧɢɯ ɮɚɤɬɨɪɿɜ (
ɯ
ɋ
) ɩɿɞɥɹɝɚɽ ɪɨɡɩɨɞɿɥɭ ɧɚ ɞɢɫɩɟɪɫɿʀ ɤɨɠɧɨɝɨ ɡ
ɮɚɤɬɨɪɿɜ ɨɤɪɟɦɨ (
ɋȼȺ
ɋɋɋ ,,
), ɚ ɬɚɤɨɠ ɞɢɫɩɟɪɫɿʀ ɪɿɡɧɢɯ ɜɚɪɿɚɧɬɿɜ ʀɯ ɫɩɨɥɭɱɟɧɶ
),,,(
ȼɋȺɋȺȼȺȼɋ
ɋɋɋɋ
. ɍ ɧɟɪɿɜɧɨɦɿɪɧɢɯ ɤɨɦɩɥɟɤɫɚɯ ɜɫɿ ɱɚɫɬɤɨɜɿ ɞɢɫɩɟɪɫɿʀ ɮɚɤɬɨɪɿɜ
ɜɿɞɪɿɡɧɹɬɢɦɭɬɶɫɹ ɜɿɞ ɜɟɥɢɱɢɧ ɬɚɤɢɯ ɠɟ ɞɢɫɩɟɪɫɿɣ ɜ ɪɿɜɧɨɦɿɪɧɨɦɭ ɤɨɦɩɥɟɤɫɿ,
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ɬɚɛɥɢɰɹɯ 51 ɿ 52.
168
169
Ɍɚɛɥɢɰɹ 50
Ɉɛɪɨɛɤɚ ɬɪɢɮɚɤɬɨɪɧɨɝɨ ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɤɨɦɩɥɟɤɫɭ
(Ⱥ – ɝɪɭɩɢ ɩɿɞɩɪɢɽɦɫɬɜ ɡɚ ɪɿɜɧɟɦ ɩɪɨɞɭɤɬɢɜɧɨɫɬɿ ɩɪɚɰɿ, ɥɸɞɢɧɨ-ɝɨɞɢɧ ɧɚ 1 ɰ ; ȼ – ɪɨɡɦɿɪ ɜɢɬɪɚɬ ɤɨɪɦɿɜ ɧɚ
ɜɢɪɨɛɧɢɰɬɜɨ 1ɰ ɹɥɨɜɢɱɢɧɢ, ɰ ɤɨɪɦ. ɨɞ.; ɋ- ɫɨɛɿɜɚɪɬɿɫɬɶ 1ɰ ɤɨɪɦɨɜɢɯ ɨɞɢɧɢɰɶ, ɡɝɨɞɨɜɚɧɢɯ ɯɭɞɨɛɿ, ɝɪɧ;
V – ɪɿɜɟɧɶ ɫɨɛɿɜɚɪɬɨɫɬɿ 0,1ɰ ɹɥɨɜɢɱɢɧɢ, ɝɪɧ.;
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Ƚɪɭɩɢ ɬɚ ɩɿɞɝɪɭɩɢ ɡɚ
ɮɚɤɬɨɪɚɦɢ
ȼɢɯɿɞɧɿ ɿ ɪɨɡɪɚɯɭɧɤɨɜɿ ɞɚɧɿ
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Ɍɚɛɥɢɰɹ 51
Ⱦɨɩɨɦɿɠɧɿ ɪɨɡɪɚɯɭɧɤɢ ɞɥɹ ɨɛɪɨɛɤɢ ɞɢɫɩɟɪɫɿɣɧɨɝɨ ɤɨɦɩɥɟɤɫɭ
ɡɚ ɮɚɤɬɨɪɚɦɢ Ⱥ, ȼ, ɋ
Ƚɪɭɩɢ ɿ
ɩɿɞɝɪɭɩɢ ɡɚ
ɮɚɤɬɨɪɚɦɢ
ɑɢɫɥɨ
ɫɟɪɟɞɧɿɯ (l)
ɑɢɫɥɨ
ɫɩɨɫɬɟɪɟɠɟɧɶ
(ɩ)
Ɋɨɡɪɚɯɭɧɤɨɜɿ ɞɚɧɿ
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4 27 366,92 8414,39 2534,63 2534,63 93,87
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4 39 419,08 104,77 10976,75 4065,14 104,23
ɉɨɤɚɡɧɢɤɢ ɩɨ
ɮɚɤɬɨɪɭ
66 786,00 - 19391,14 6599,77 100,00
Ⱥ
1
ȼ
4 32 363,47 90,87 8257,36 301,99 91,19
2
ȼ
4 34 422,53 105,63 11157,70 3585,78 105,46
ɉɨɤɚɡɧɢɤɢ ɩɨ
ɮɚɤɬɨɪɭ
ȼ
66 786,00 - 19415,06 6599,77 100,00
1
ɋ
4 33 353,03 88,26 7789,83 3031,59 91,87
2
ɋ
4 38 432,97 108,24 11715,90 3568,18 108,18
ɉɨɤɚɡɧɢɤɢ ɩɨ
ɮɚɤɬɨɪɭ ɋ
66 786,00 - 19505,73 6599,77 100,0
170