Материал: chap5

n2 = 144, Ɇȼ2 = 81,6, sȼ2 =2,7,

Ʉɚɤ ɜɢɞɢɦ, ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ t ɧɚɦɧɨɝɨ ɩɪɟɜɵɲɚɟɬ ɬɪɟɛɭɟɦɨɟ ɞɥɹ 1%-ɝɨ ɭɪɨɜɧɹ (2,58), ɬ.ɟ. ɫɜɹɡɶ ɟɫɬɶ.

ɍɩɪɚɠɧɟɧɢɟ 91. ɉɨ ɩɪɢɜɟɞɟɧɧɵɦ ɜ ɩɪɢɦɟɪɟ ɞɚɧɧɵɦ ɩɪɨɜɟɪɢɬɶ ɧɚ 1%-ɧɨɦ ɭɪɨɜɧɟ ɡɧɚɱɢɦɨɫɬɶ ɪɚɡɥɢɱɢɣ ɫɪɟɞɧɟɣ ɡɚɪɩɥɚɬɵ ɧɟɜɟɫɬ ɞɥɹ ɞɜɭɯ ɝɪɭɩɩ ɠɟɧɢɯɨɜ: ɫ ɡɚɪɩɥɚɬɨɣ 50100

ɪɭɛ. ɢ 100—150 ɪɭɛ.

Ɉɬɜɟɬ: t = 12,1; v = 290, ɪɚɡɥɢɱɢɟ ɡɧɚɱɢɦɨ.

7. Ɂɧɚɱɢɦɨɫɬɶ ɪɚɡɥɢɱɢɣ ɞɢɫɩɟɪɫɢɢ

ɂɡ ɞɜɭɯ ɛɟɫɤɨɧɟɱɧɵɯ ɧɨɪɦɚɥɶɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɵɯ ɝɟɧɟɪɚɥɶɧɵɯ ɫɨɜɨɤɭɩɧɨɫɬɟɣ (ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɨ ɧɨɪɦɚɥɶɧɨɫɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɡɞɟɫɶ ɫɭɳɟɫɬɜɟɧɧɨ, ɟɫɥɢ ɢɫɫɥɟɞɨɜɚɬɟɥɶ ɫɨɦɧɟɜɚɟɬɫɹ ɜ ɟɝɨ ɜɟɪɧɨɫɬɢ, ɫɥɟɞɭɟɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɪɭɝɢɟ ɦɟɬɨɞɵ28) ɢɡɜɥɟɱɟɧɵ ɧɟɡɚɜɢɫɢɦɵɟ ɜɵɛɨɪɤɢ ɨɛɴɟɦɚ ɩ1 ɢ n2. Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ, ɪɚɜɧɵ ɥɢ ɞɢɫɩɟɪɫɢɢ ɝɟɧɟɪɚɥɶɧɵɯ ɫɨɜɨɤɭɩɧɨɫɬɟɣ.

1. H0 : V1Ƚ 2 V2Ƚ 2 H1 : V1Ƚ 2 z V2Ƚ 2

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2. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ V1B 2 ɛɨɥɶɲɭɸ ɢɡ ɜɵɛɨɪɨɱɧɵɯ ɞɢɫɩɟɪɫɢɣ. Ⱦɥɹ ɩɪɨɜɟɪɤɢ ɝɢɩɨɬɟɡɵ ɇ0 ɩɪɨɬɢɜ ɇ1 ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɨɬɧɨɲɟɧɢɟ ɜɵɛɨɪɨɱɧɵɯ ɞɢɫɩɟɪɫɢɣ ɢɥɢ ɨɰɟɧɨɤ (ɞɟɥɨ ɜ ɬɨɦ, ɱɬɨ

Jɬɧɨɲɟɧɢɟ ɞɢɫɩɟɪɫɢɣ F ɢɦɟɟɬ ɬɚɤ ɧɚɡɵɜɚɟɦɨɟ F-ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫ (n1 — 1) ɢ (n2 — 1) ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ (ɟɫɥɢ ɜɟɪɧɚ ɝɢɩɨɬɟɡɚ ɇ0).

3. Ʉɪɢɬɢɱɟɫɤɢɟ ɬɨɱɤɢ ɞɥɹ ɡɚɞɚɧɧɨɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɬɚɤ: ɜɟɪɯɧɹɹ ɤɪɢɬɢɱɟɫɤɚɹ ɬɨɱɤɚ FB – ɩɨ ɫɩɟɰɢɚɥɶɧɵɦ ɬɚɛɥɢɰɚɦ (ɫɦ. ɉɪɢɥɨɠɟɧɢɟ 3, ɬɚɛɥ. Ʌ), ɚ ɧɢɠɧɹɹ Fɇ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ:

FB

Ƚɢɩɨɬɟɡɚ ɩɪɢɧɢɦɚɟɬɫɹ, ɟɫɥɢ ɪɚɫɫɱɢɬɚɧɧɨɟ ɩɨ ɮɨɪɦɭɥɟ (V,7,1) ɡɧɚɱɟɧɢɟ F ɥɟɠɢɬ ɦɟɠɞɭ FH ɢ FB, ɬ.ɟ. FH < F<FB . ȿɫɥɢ ɠɟ F < FH ɢɥɢ F > FB , ɬɨ ɝɢɩɨɬɟɡɚ ɨɬɜɟɪɝɚɟɬɫɹ ɧɚ ɡɚɞɚɧɧɨɦ ɢɫɫɥɟɞɨɜɚɬɟɥɟɦ ɭɪɨɜɧɟ ɡɧɚɱɢɦɨɫɬɢ.

4. Ⱦɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɧɚɣɞɟɧɧɵɯ ɜɵɲɟ ɤɪɢɬɢɱɟɫɤɢɯ ɬɨɱɟɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:

28 Ɂɚɤɫ Ʌ. ɋɬɚɬɢɫɬɢɱɟɫɤɨɟ ɨɰɟɧɢɜɚɧɢɟ. Ɇ., 1976, ɫ. 242, 243. Ƚɥɚɫc Ⱦɠ., ɋɬɷɧɥɢ Ⱦɠ. ɋɬɚɬɢɫɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɜ ɩɟɞɚɝɨɝɢɤɟ ɢ ɩɫɢɯɨɥɨɝɢɢ. Ɇ., 1976.

137

ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ ɛɨɥɶɲɟɣ ɞɢɫɩɟɪɫɢɢ.

ɍɩɪɚɠɧɟɧɢɟ 92.

ɉɭɫɬɶ n1 = 60, s12 = 10, ɩ2, == 140, s22 == 5. Ɉɩɪɟɞɟɥɢɬɶ ɡɧɚɱɢɦɨɫɬɶ ɪɚɡɥɢɱɢɣ ɧɚ ɭɪɨɜɧɟ 5- ɢ 95% -ɢ ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ ɨɬɧɨɲɟɧɢɹ ɝɟɧɟɪɚɥɶɧɵɯ ɞɢɫɩɟɪɫɢɣ. Ɉɬɜɟɬ: F = 2, FB

= 1,43, FH= 0,70, ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ (1,40; 2,86).

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8. Ɂɧɚɱɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɤɨɪɪɟɥɹɰɢɢ r, U, W ɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ, ɨɫɧɨɜɚɧɧɵɯ ɧɚ

F2

Ⱥ. Ʉɨɷɮɮɢɰɢɟɧɬ r

1. Ɍɪɟɛɭɟɬɫɹ ɩɪɨɜɟɪɢɬɶ ɡɧɚɱɢɦɨɫɬɶ r, ɬ.ɟ. ɦɨɠɟɬ ɥɢ ɩɪɢ ɞɚɧɧɨɦ ɡɧɚɱɟɧɢɢ ɜɵɛɨɪɨɱɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ rȼ ɛɵɬɶ ɪɚɜɧɵɦ ɧɭɥɸ ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ rȽ ɞɥɹ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ.

1.H0: rȽ=0

2.H1: rȽ z 0

2. ɋɬɚɬɢɫɬɢɤɚ t, ɪɚɫɫɱɢɬɵɜɚɟɦɚɹ ɩɨ ɮɨɪɦɭɥɟ (V,8,1) ɢɦɟɟɬ t-ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫ (ɩ—2) ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ:

ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ q, ɩɪɢ t t tɤɪ ɝɢɩɨɬɟɡɚ ɇ0 : r Ƚ = 0 ɨɬɤɥɨɧɹɟɬɫɹ ɧɚ ɭɪɨɜɧɟ ɡɧɚɱɢɦɨɫɬɢ q.

4. Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɞɨɜɟɪɢɬɟɥɶɧɵɯ ɢɧɬɟɪɜɚɥɨɜ ɜɵɛɨɪɨɱɧɨɟ ɡɧɚɱɟɧɢɟ rȼ ɩɨɞɜɟɪɝɚɟɬɫɹ ɬɚɤ ɧɚɡɵɜɚɟɦɨɦɭ ɩɪɟɨɛɪɚɡɨɜɚɧɢɸ Ɏɢɲɟɪɚ:

z 1 ln1 r (V8.2)

21 r

Ⱦɟɥɨ ɜ ɬɨɦ, ɱɬɨ rB ɢɦɟɟɬ ɧɨɪɦɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɥɢɲɶ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ rȽ = 0, ɚ ɜɨɬ zȼ

— ɩɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ zȽ. ɉɨɷɬɨɦɭ, ɪɚɫɫɱɢɬɚɜ zȼ ɩɨ ɩɨɥɭɱɟɧɧɨɦɭ ɜ ɜɵɛɨɪɤɟ ɡɧɚɱɟɧɢɸ rB, ɫɬɪɨɢɦ ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ rȽ (z ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɬɚɛɥɢɰɟ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ Ⱥ ɉɪɢɥɨɠɟɧɢɹ 3, ɧɚɩɪɢɦɟɪ ɞɥɹ 95% ɞɨɜɟɪɢɬɟɥɶɧɨɝɨ ɢɧɬɟɪɜɚɥɚ z =1,96):

ɉɨɥɭɱɢɜ ɧɢɠɧɟɟ ɢ ɜɟɪɯɧɟɟ ɡɧɚɱɟɧɢɹ z, ɪɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɹ rɧɢɠɧ ɂ rɜɟɪɯɧ ɩɨ ɮɨɪɦɭɥɟ

re2z 1

e2z 1 (V.8.3)

ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ (V,8,3) ɦɨɠɧɨ ɬɚɤɠɟ ɨɫɭɳɟɫɬɜɢɬɶ ɩɨ ɬɚɛɥ. Ʉ ɉɪɢɥɨɠɟɧɢɹ 3.

29 Ɂɧɚɱɢɦɨɫɬɶ r ɦɨɠɧɨ ɨɰɟɧɢɬɶ ɬɚɤɠɟ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɨ ɬɚɛɥɢɰɟ ȿ ɉɪɢɥɨɠɟɧɢɹ 3 (ɛɟɡ ɩɪɨɜɟɞɟɧɢɹ ɤɚɤɢɯ-ɥɢɛɨ ɪɚɫɱɟɬɨɜ).

138

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5. Ɋɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɢɣ ɩɪɢɦɟɪ 32. ȼ ɢɫɫɥɟɞɨɜɚɧɢɢ ɢɧɠɟɧɟɪɨɜ ɥɟɧɢɧɝɪɚɞɫɤɢɯ ɩɪɨɟɤɬɧɨɤɨɧɫɬɪɭɤɬɨɪɫɤɢɯ ɨɪɝɚɧɢɡɚɰɢɣ ɛɵɥɢ ɩɨɥɭɱɟɧɵ ɞɚɧɧɵɟ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟ ɫɜɹɡɶ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɟɣ ɩɪɨɮɟɫɫɢɟɣ ɢ ɪɚɛɨɬɨɣ30. Ʉɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ, ɪɚɫɫɱɢɬɚɧɧɵɣ ɩɨ ɨɰɟɧɤɚɦ 89 ɪɭɤɨɜɨɞɢɬɟɥɟɣ ɝɪɭɩɩ, ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɪɚɛɨɬɨɣ ɢ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɩɪɨɮɟɫɫɢɟɣ ɪɚɜɟɧ 0,23. ɉɪɨɜɟɪɢɦ ɡɧɚɱɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɚ ɧɚ 1%-ɧɨɦ ɭɪɨɜɧɟ ɢ ɩɨɫɬɪɨɢɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ (ɬ.ɟ. 99%-ɣ) ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ.

ɂɬɚɤ, ɩ = 89, rB = 0,23. ɉɨ ɮɨɪɦɭɥɟ (V,8,1) ɩɨɥɭɱɚɟɦ:

ɉɨ ɬɚɛɥ. ɂ ɉɪɢɥɨɠɟɧɢɹ 3 ɧɚɯɨɞɢɦ, ɱɬɨ ɞɥɹ ɩ — 2 = 87 ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ 1%-ɹ ɤɪɢɬɢɱɟɫɤɚɹ ɬɨɱɤɚ ɪɚɜɧɚ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ 2,64 (ɜ ɬɚɛɥɢɰɟ ɧɟ ɩɪɢɜɟɞɟɧɵ ɤɪɢɬɢɱɟɫɤɢɟ ɡɧɚɱɟɧɢɹ ɞɥɹ 87 ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ, ɧɨ ɩɪɢɜɟɞɟɧɵ ɞɥɹ 60, ɪɚɜɧɵɟ 2,66 ɢ ɞɥɹ 120 ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ — 2,62, ɬ.ɟ. ɢɫɤɨɦɨɟ ɡɧɚɱɟɧɢɟ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ɥɟɠɢɬ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɩɨɫɪɟɞɢɧɟ ɦɟɠɞɭ 2,62 ɢ 2,66). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɯɨɬɹ ɜ ɰɢɬɢɪɨɜɚɧɧɨɣ ɤɧɢɝɟ, ɨɬɤɭɞɚ ɜɡɹɬ ɷɬɨɬ ɩɪɢɦɟɪ, ɭɤɚɡɚɧɨ, ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬ ɡɧɚɱɢɦ ɧɚ ɭɪɨɜɧɟ 1%, ɨɧ ɡɧɚɱɢɦ ɥɢɲɶ ɧɚ 5%-ɧɨɦ ɭɪɨɜɧɟ (ɤɚɤ ɜɢɞɧɨ ɢɡ ɬɚɛɥɢɰɵ, ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɞɥɹ ɭɪɨɜɧɹ 5% tɤɪ =1,99). Ɍɚɤɨɣ ɠɟ ɪɟɡɭɥɶɬɚɬ ɞɚɟɬ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɬɚɛɥɢɰɵ ȿ ɉɪɢɥɨɠɟɧɢɹ 3.

ɉɨɫɬɪɨɢɦ ɬɟɩɟɪɶ 99%-ɢ ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ. ɉɨ ɮɨɪɦɭɥɟ (V,8,2) ɩɨɥɭɱɚɟɦ: zB 1 ln1 0,23 | 0,234

21 0,23

ɉɨ ɬɚɛɥɢɰɟ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɚɯɨɞɢɦ, ɱɬɨ ɞɥɹ 1%-ɝɨ ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, —0,045 < zȽ < 0,513. ɇɨ ɷɬɨ ɞɨɜɟɪɢɬɟɥɶɧɵɟ ɝɪɚɧɢɰɵ ɞɥɹ z. Ɍɟɩɟɪɶ ɧɟɨɛɯɨɞɢɦɨ ɨɩɹɬɶ ɜɟɪɧɭɬɶɫɹ ɤ ɤɨɷɮɮɢɰɢɟɧɬɚɦ ɤɨɪɪɟɥɹɰɢɢ. ɉɨ ɮɨɪɦɭɥɟ (V,8,3) ɩɨɥɭɱɚɟɦ ɞɥɹ

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ɧɢɠɧɟɣ ɝɪɚɧɢɰɵ:

e 2 0,045 1

e 2 0,045 1

0,04

Ⱦɥɹ ɜɟɪɯɧɟɣ ɝɪɚɧɢɰɵ ɚɧɚɥɨɝɢɱɧɵɟ ɪɚɫɱɟɬɵ ɞɚɸɬ 0,47.

ȼɦɟɫɬɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨ ɮɨɪɦɭɥɚɦ (V,8,2) ɢ (V,8,3) ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɬɨɬ ɠɟ ɪɟɡɭɥɶɬɚɬ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɩɨ ɬɚɛɥɢɰɟ Ʉ ɉɪɢɥɨɠɟɧɢɹ 3. ɇɚɩɪɢɦɟɪ, ɞɥɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ 0,23 ɧɚɯɨɞɢɦ ɫɬɪɨɤɭ 0,2 ɢ ɫɬɨɥɛɟɰ 3 — ɧɚ ɢɯ ɩɟɪɟɫɟɱɟɧɢɢ ɫɬɨɢɬ z, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ r = 0,23, ɚ ɢɦɟɧɧɨ 0,2342. Ⱥ ɱɬɨɛɵ ɩɪɟɨɛɪɚɡɨɜɚɬɶ z = 0,513, ɧɚɯɨɞɢɦ ɜɧɭɬɪɢ ɬɚɛɥɢɰɵ ɧɚɢɛɨɥɟɟ ɛɥɢɡɤɨɟ ɤ ɧɟɦɭ ɱɢɫɥɨ (ɷɬɨ 0,5101). Ɉɧɨ ɫɬɨɢɬ ɜ ɫɬɪɨɤɟ 0,4 ɢ ɫɬɨɥɛɰɟ 7, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, r = 0,47. ɂɬɚɤ, 99% ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɪɚɜɟɧ:

— 0,04 < rȽ < 0,47,

ɉɨ ɬɚɛɥɢɰɟ ɀ ɉɪɢɥɨɠɟɧɢɹ 3 ɦɨɠɧɨ ɧɚɣɬɢ, ɧɟ ɩɪɨɜɨɞɹ ɪɚɫɱɟɬɨɜ, ɩɪɢɛɥɢɠɟɧɧɨ 95%-ɣ

ɢɧɬɟɪɜɚɥ: 0,03 < rȽ < 0,42.

30 ɋɨɰɢɚɥɶɧɨ-ɩɫɢɯɨɥɨɝɢɱɟɫɤɢɣ ɩɨɪɬɪɟɬ ɢɧɠɟɧɟɪɚ. Ɇ., 1977, ɬɚɛɥ. 11 ɧɚ ɫ. 149.

139

6. ɍɩɪɚɠɧɟɧɢɟ 93. ȼ ɬɨɣ ɠɟ ɪɚɛɨɬɟ31 ɭɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ, ɩɨɥɭɱɟɧɧɵɣ ɞɥɹ 52 ɝɥɚɜɧɵɯ ɢɧɠɟɧɟɪɨɜ ɩɪɨɟɤɬɚ ɢ ɪɚɜɧɵɣ 0,51, ɡɧɚɱɢɦ ɧɚ ɭɪɨɜɧɟ 1%. ɉɪɨɜɟɪɢɬɶ, ɬɚɤ ɥɢ ɷɬɨ, ɢ ɩɨɫɬɪɨɢɬɶ ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ. Ɉɬɜɟɬ: ɤɨɷɮɮɢɰɢɟɧɬ ɡɧɚɱɢɦ (t = 4,19); 0,19 < rȽ < 0,73.

Ȼ. Ʉɨɷɮɮɢɰɢɟɧɬ ɪɚɧɝɨɜɨɣ ɤɨɪɪɟɥɹɰɢɢ U

1.H0: UȽ=0

2.H1: UȽ z 0

2. ȿɫɥɢ ɇ0 ɜɟɪɧɚ ɢ ɩ > 10 (ɩɪɢ ɩ d10 ɡɧɚɱɢɦɨɫɬɶ U ɨɩɪɟɞɟɥɹɟɬɫɹ ɞɪɭɝɢɦ ɫɩɨɫɨɛɨɦ ɩɨ ɫɩɟɰɢɚɥɶɧɨɣ ɬɚɛɥɢɰɟ32, ɬɨ ɡɧɚɱɟɧɢɟ t, ɪɚɫɫɱɢɬɚɧɧɨɟ ɩɨ ɮɨɪɦɭɥɟ (V,8,4), ɢɦɟɟɬ t — ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɋɬɶɸɞɟɧɬɚ ɫ (ɩ — 2) ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ:

ɜɟɧɧɨ ɩɨ ɡɧɚɱɟɧɢɸ U (ɛɟɡ ɪɚɫɱɟɬɚ t) ɩɨ ɬɚɛɥɢɰɟ ȼ ɉɪɢɥɨɠɟɧɢɹ 3.

4. Ⱦɨɜɟɪɢɬɟɥɶɧɵɟ ɢɧɬɟɪɜɚɥɵ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ U ɧɟ ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ, ɬɚɤ ɤɚɤ,

ɨɤɚɡɵɜɚɟɬɫɹ, ɩɨɥɭɱɢɬɶ ɜɵɛɨɪɨɱɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɞɥɹ UB ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ UȽ z 0, ɨɱɟɧɶ ɫɥɨɠɧɨ33.

ɉɪɢɦɟɪ 33. ȼ ɩɪɢɦɟɪɟ 19 ɛɵɥ ɪɚɫɫɱɢɬɚɧ ɤɨɷɮɮɢɰɢɟɧɬ ɫɜɹɡɢ ɦɟɠɞɭ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɨɬɜɟɬɚɦɢ ɧɚ ɜɨɩɪɨɫɵ «ɢɧɬɟɪɟɫɧɚɹ ɪɚɛɨɬɚ» ɢ «ɨɛɪɚɡɨɜɚɧɢɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɚɛɨɬɟ» ɞɥɹ 14 ɝɪɭɩɩ ɪɚɛɨɱɢɯ. Ɉɤɚɡɚɥɨɫɶ, ɱɬɨ UB=0,345. Ɉɩɪɟɞɟɥɢɦ ɧɚ 5%-ɧɨɦ ɭɪɨɜɧɟ ɡɧɚɱɢɦɨɫɬɶ ɷɬɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ.

>1 0,345 2 @ 1 12

ɉɨɫɤɨɥɶɤɭ ɷɬɨ ɦɟɧɶɲɟ ɤɪɢɬɢɱɟɫɤɨɝɨ ɡɧɚɱɟɧɢɹ 2,23 (ɞɥɹ 10 ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ, ɬɚɤ ɤɚɤ ɜ ɬɚɛɥɢɰɟ ɧɟ ɩɪɢɜɟɞɟɧɵ ɡɧɚɱɟɧɢɹ ɞɥɹ 12), ɤɨɷɮɮɢɰɢɟɧɬ ɧɟɡɧɚɱɢɦ, ɯɨɬɹ ɜ ɰɢɬɢɪɨɜɚɧɧɨɣ ɪɚɛɨɬɟ ɨɧ ɢɧɬɟɪɩɪɟɬɢɪɭɟɬɫɹ ɤɚɤ ɡɧɚɱɢɦɵɣ.

ɍɩɪɚɠɧɟɧɢɟ 94. Ɉɩɪɟɞɟɥɢɬɶ, ɛɭɞɟɬ ɥɢ ɡɧɚɱɢɦ ɤɨɷɮɮɢɰɢɟɧɬ, ɟɫɥɢ ɨɧ ɪɚɫɫɱɢɬɚɧ ɞɥɹ 42 ɝɪɭɩɩ. Ɉɬɜɟɬ: Ⱦɚ (t = 2,32).

ȼ. Ʉɨɷɮɮɢɰɢɟɧɬ ɪɚɧɝɨɜɨɣ ɤɨɪɪɟɥɹɰɢɢ W

ȼɨɩɪɨɫ ɨ ɫɭɳɟɫɬɜɟɧɧɨɫɬɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬ ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɥɢ ɪɚɧɟɟ (§ 6, ɝɥ. II), ɬɚɦ ɠɟ ɩɨɤɚɡɚɧɨ, ɤɚɤɢɦ ɨɛɪɚɡɨɦ ɨɩɪɟɞɟɥɹɬɶ ɡɧɚɱɢɦɨɫɬɶ W ɩɪɢ ɩ d 10 (ɩɨɫɤɨɥɶɤɭ ɞɥɹ ɩɪɨɜɟɪɤɢ ɫɭɳɟɫɬɜɟɧɧɨɫɬɢ ɢɫɩɨɥɶɡɭɟɬɫɹ S, ɦɵ ɫɨɱɥɢ ɰɟɥɟɫɨɨɛɪɚɡɧɵɦ ɪɚɫɫɦɨɬɪɟɬɶ ɷɬɨɬ ɜɨɩɪɨɫ ɫɪɚɡɭ ɩɨɫɥɟ ɜɜɟɞɟɧɢɹ S). ɉɭɫɬɶ ɩ > 10.

1.H0: WȽ=0

2.H1: WȽ z 0

2. ȿɫɥɢ ɇ0 ɜɟɪɧɚ, ɜɟɥɢɱɢɧɚ z ɢɦɟɟɬ ɧɨɪɦɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ:

31ɋɨɰɢɚɥɶɧɨ-ɩɫɢɯɨɥɨɝɢɱɟɫɤɢɣ ɩɨɪɬɪɟɬ ɢɧɠɟɧɟɪɚ. Ɇ., 1977, ɫ. 149.

32Ʉɟɧɞɷɥ Ɇ. Ɋɚɧɝɨɜɵɟ ɤɨɪɪɟɥɹɰɢɢ. Ɇ., 1975, ɫ. 69, 188—191.

33Ʉɟɧɞɷɥ Ɇ. Ɋɚɧɝɨɜɵɟ ɤɨɪɪɟɥɹɰɢɢ. Ɇ., 1975, ɫ. 102. ɋɦ. ɬɚɤɠɟ Ʉɟɧɞɚɥɥ Ɇ. Ⱦɠ., ɋɬɶɸɚɪɬ Ⱥ. ɋɬɚɬɢɫɬɢɱɟɫɤɢɟ ɜɵɜɨɞɵ ɢ ɫɜɹɡɢ. Ɇ., 1973, ɫ. 637—644.

140

S* S 1, ɟɫɥɢ S>0

zS * , ɝɞɟ S*=0, ɟɫɥɢ S=0; (VI,8,5)

V

S* S 1, ɟɫɥɢ S<0

V1 n(n 1)(2n 5),

18

[202]

ɟɫɥɢ ɧɟɬ ɨɛɴɟɞɢɧɟɧɧɵɯ ɪɚɧɨɝɨɜ ɢ

ɟɫɥɢ ɟɫɬɶ ɨɛɴɟɞɢɧɟɧɧɵɟ ɪɚɧɝɢ; V, ɪɚɡɭɦɟɟɬɫɹ, ɪɚɜɧɚ ɤɨɪɧɸ ɤɜɚɞɪɚɬɧɨɦɭ ɢɡ ɩɪɢɜɟɞɟɧɧɨɝɨ ɜɵɪɚɠɟɧɢɹ; tr ɢ ɢs — ɱɢɫɥɨ ɨɛɴɟɞɢɧɟɧɧɵɯ ɪɚɧɝɨɜ ɜ r-ɦ ɨɛɴɟɞɢɧɟɧɢɢ ɩɨ ɏ ɢ s-ɦ ɨɛɴɟɞɢɧɟɧɢɢ ɩɨ Y ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.

3.Ʉɪɢɬɢɱɟɫɤɢɟ ɬɨɱɤɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɬɚɛɥɢɰɟ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɇ0 ɨɬɜɟɪɝɚɟɬɫɹ ɩɪɢ | z| > zɤɪ. ɉɪɢ ɨɬɫɭɬɫɬɜɢɢ ɨɛɴɟɞɢɧɟɧɧɵɯ ɪɚɧɝɨɜ ɡɧɚɱɢɦɨɫɬɶ W ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɨ ɬɚɛɥɢɰɟ Ⱦ ɉɪɢɥɨɠɟɧɢɹ 3 (ɛɟɡ ɪɚɫɱɟɬɚ ɝ).

4.Ⱦɨɜɟɪɢɬɟɥɶɧɵɟ ɢɧɬɟɪɜɚɥɵ ɞɥɹ W ɧɟ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɬɟɯ ɠɟ ɫɨɨɛɪɚɠɟɧɢɣ, ɱɬɨ ɢ ɞɨɜɟɪɢɬɟɥɶɧɵɟ ɢɧɬɟɪɜɚɥɵ ɞɥɹ U.

5.ɉɪɢɦɟɪ 34. ɉɭɫɬɶ ɩ = 20, S = 52 (W = 0,27). Ɉɩɪɟɞɟɥɢɦ ɡɧɚɱɢɦɨɫɬɶ W ɧɚ ɭɪɨɜɧɟ 5%. Ɍɚɤ ɤɚɤ S ɩɨɥɨɠɢɬɟɥɶɧɨ, S* ɪɚɜɧɨ 51.

Ɍɚɤ ɤɚɤ ɞɥɹ 5%-ɧɨɝɨ ɭɪɨɜɧɹ ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɪɚɜɧɨ 1,96 (ɬɚɛɥ Ⱥ ɉɪɢɥɨɠɟɧɢɹ 3), ɝɢɩɨɬɟɡɚ ɇ0 ɩɪɢɧɢɦɚɟɬɫɹ, ɤɨɷɮɮɢɰɢɟɧɬ ɧɟɡɧɚɱɢɦ.

ɍɩɪɚɠɧɟɧɢɟ 95. S = 33 (W = 0,36), ɩ = 14. ɇɚɣɬɢ, ɡɧɚɱɢɦ ɥɢ ɤɨɷɮɮɢɰɢɟɧɬ ɧɚ ɭɪɨɜɧɟ 5%. Ɉɬɜɟɬ: ɧɟɡɧɚɱɢɦ, z=1,75.

Ƚ. Ʉɨɷɮɮɢɰɢɟɧɬɵ, ɨɫɧɨɜɚɧɧɵɟ ɧɚ F2

Ʉɚɤ ɭɠɟ ɭɤɚɡɵɜɚɥɨɫɶ, ɫɭɳɟɫɬɜɟɧɧɨɫɬɶ ɢɯ ɩɪɨɜɟɪɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ F2. ɟɫɥɢ ɡɧɚɱɢɦ F2, ɬɨ ɡɧɚɱɢɦ ɢ ɪɚɫɫɱɢɬɚɧɧɵɣ ɫ ɟɝɨ ɩɨɦɨɳɶɸ ɤɨɷɮɮɢɰɢɟɧɬ. ɉɨɷɬɨɦɭ ɩɪɢ ɪɚɫɱɟɬɚɯ, ɤɨɬɨɪɵɟ ɞɥɹ ɬɚɛɥɢɰ k ul , ɤɚɤ ɩɪɚɜɢɥɨ, ɩɪɨɢɡɜɨɞɹɬɫɹ ɧɚ ɗȼɆ, ɠɟɥɚɬɟɥɶɧɨ ɜɵɩɟɱɚɬɵɜɚɬɶ ɧɟ ɬɨɥɶɤɨ ɡɧɚɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ,

[203]

ɧɨ ɢ ɡɧɚɱɟɧɢɟ F2. ȿɫɥɢ ɷɬɨ ɧɟ ɫɞɟɥɚɧɨ, ɬɨ F2 ɦɨɠɧɨ, ɪɚɡɭɦɟɟɬɫɹ, ɥɟɝɤɨ ɧɚɣɬɢ, ɩɪɟɨɛɪɚɡɨɜɚɜ ɮɨɪɦɭɥɵ ɞɥɹ ɪɚɫɱɟɬɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ. ɇɚɩɪɢɦɟɪ, ɩɪɟɨɛɪɚɡɨɜɚɜ ɮɨɪɦɭɥɭ ɞɥɹ ɪɚɫɱɟɬɚ ɤɨɷɮɮɢɰɢɟɧɬɚ ɑɭɩɪɨɜɚ Ɍ, ɩɨɥɭɱɢɦ:

F2 T 2 n( k 1)( l 1)

141