Сб задач по МС Каштанова 1 12 2010
.pdf, ,
,
,
.
: 1. .
|
. |
|
. |
2. |
. |
3. |
. |
4. |
. |
.
1. .
,
0, |
. . |
, |
0 |
;
,
0 .
2. |
. |
,
. 2 .
4.1. (
)
t-: X1,..,Xn
2 .
H0 : 0 .
H1 : 0 H1 : 0 (0 )
: .
:
1. |
, |
0 |
1, .
2. X1,..,Xn n.
3.:
|
|
x |
0 |
|
|
|
T |
|
|
n 1 , |
|||
|
|
S |
|
|||
|
|
|
|
|
|
, S x2 - X1,..,Xn,.
t- = n 1 .
31
1. , 400 . 26 380
50 .
|
0,05. |
|
|
2. |
7 |
30 |
. |
|
|
|
: |
29, 31, 30, 27, 25, 34, 36, 35, 29, 30, 28, 42, 31, 29, 29, 32, 27 ( |
). |
|
|
|
, |
|
|
0,05?
|
3. |
, |
|
|
, |
6 |
. |
12 |
|
|
8 |
1 |
. |
|
|
5,6 |
2. |
|
|
, |
|
|
0,05? |
|
4. |
|
|
|
|
|
400 |
. |
|
|
. |
|
|
|
|
|
. |
|
|
|
. |
|
9 |
|
. |
2. |
|
|
388 |
|
171 |
|
|
, |
|
|
|
0,01? |
|
|
|
|
5. |
|
|
|
|
|
. |
|
12 |
|
, |
|
200 |
MN . |
|
|
: 198, 202, 200, |
204, 196, 190, 192, 201, 200, 196, 196, 190. |
|
||||
0,05, |
|
|
, |
|
. |
|
6. |
|
|
|
-180 |
|
13 |
. |
|
|
. |
|
|
|
. |
|
|
|
|
|
|
8 |
. |
|
|
14,6 |
S2 = 5,12 |
2. |
|
0,01 |
|
|
|
|
. |
|
7. |
|
, |
|
|
|
|
|
40 |
, |
36. |
|
|
|
2. |
|
40,2 |
|
|
1,2 |
|
|
,
? = 0,01.
8. 6 (§2) , 610 . 0,05.
9. 9 (§2) , 1 .
0,05.
33
4.2. (
)
t-: X1,..,Xn Y1,..,Y
x y
2 , n, m < 30.
H 0 : x y .
H1 : x |
y |
H1 : x y ( x y ) |
, |
: ,
.
: .
:
1. , H 0 H1 , .
2. X1,..,Xn Y1,..,Y n m.
3.:
|
|
|
|
T |
|
|
x |
|
y |
|
|
|
nm(n |
m 2) |
, |
|
|
|
|
|
|
|
|
|
|
||||||
|
|
|
|
|
|
nSx2 |
mS y2 |
|
|
n |
m |
||||
|
|
|
|
, S x2 - |
|
|
|
|
|
|
|
|
|
X1,..,Xn, |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
, S y2 - |
|
|
|
|
|
|
|
Y1,..,Y . |
|||||
y |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
t- |
|
|
|
|
= n+m 2 |
|
. |
|
|
|
4. |
|
|
|
|
|
|
|
|
|
|
. |
|
|
|
|
|
|
) H1 : |
|
x |
y t =t |
( ; ). |
t =t |
( ; ) |
|
|
( |
|
4) |
|
|
|
, |
|
|
|
|
( |
|
|
|
). |
| T| > t |
, |
H 0 |
|
|
. |
|
|
|
|
|
) H1 : |
x |
y |
t =t |
( ; ). |
T> t , |
|
H 0 |
. |
) H1 : |
x |
y |
t = t |
=-t |
( ; ). |
T< t |
, |
H 0 |
.
|
t =t |
( ; ), t = t |
=-t |
( ; ) |
( |
4) |
|
|
, |
|
( |
|
|
). |
34
: |
H 0 |
, |
|
|
. |
|
|
. |
|
. |
, |
|
. |
15 |
17 (). ()
6,2% 1,12. , ,
() , 5,3% 0,85. 0,05.
. t-.
1. |
, |
.
H0 : 1 2 .
H1 : 1 2 .
|
|
|
|
|
|
|
: = 0,05 . |
|
|
|
|
|
|||||
2.-3. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
: |
|
|
|
|
6,2, |
Sx2 |
1,12, |
|
n |
15, |
|
|
|
|
|
|||
x |
|
|
|
|
|
||||||||||||
|
|
|
|
|
5,3, |
S y2 |
0,85, |
|
m |
17, |
|
|
|
|
|
||
|
|
y |
|
|
|
|
|
|
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
t |
|
|
|
6,2 |
5,3 |
|
|
|
15 17(15 |
17 |
2) |
2.62 |
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
15 |
17 |
|
|
||||
15 1,12 |
17 |
0,85 |
|
|
|
|
|
|
|
|
|||||||
= 15 + 17 2 = 30. |
|
|
|
|
|
|
|
|
|||||||||
4. |
|
|
|
|
|
|
|
|
|
|
|
|
|
( |
4) |
||
t =t |
|
( ; )= t |
(0,05; 30)=1,7. |
|
|
|
|
|
|||||||||
: |
|
|
|
|
| T| > |
t |
(2,62>1,7), |
|
|
H 0 |
, . . |
.
§ 4.2.
|
|
, |
|
|
|
|
|
, |
|
|
. |
|
|
1. |
|
|
|
|
|
(12 |
10 |
) |
- |
|
|
|
, |
|
|
20,5 |
, |
|
18,5 |
, |
|
|
, |
|
0,7 |
0,8. |
|
|
|
0,01 |
|
|
|
|
|
. |
|
|
|
|
|
2. |
|
2 |
|
|
. |
|
|
|
, |
|
|
|
|
. 8 ,
6 . 7,42 0,057 2, 7,63 0,0457 2 .
35
1%, ,
?
3. 2 : 16 12 . , .
30 17
2, 32 41 2.
? 0,10.
4. |
: |
, ()
. |
, |
,
, 9
.
12,5
36,5 2, 11,5 42,1 2.
, |
5%, |
?
5.
.
.
: 76,10; 76,20; 76,00; 76,04; 76,10; 76,08; 76,18; 76,02.
: 76,20; 76,00; 76,25; 76,02; 76,18; 76,06; 76,09; 76,25.
0,01
.
6. |
|
|
|
|
|
|
|
|
|
|
, |
|
|
|
|
|
|
, |
|
|
|
|
|
|
|
|
|
|
|
. |
|
|
|
|
|
|
|
|
|
|
|
, |
||
|
|
|
|
|
|
|
. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
|
|
|
|
|
( |
) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
39 |
42 |
28 |
35 |
38 |
30 |
|
30 |
|
29 |
36 |
|
34 |
|
|
28 |
31 |
29 |
30 |
24 |
37 |
|
27 |
|
22 |
33 |
|
31 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
5%. |
|
|
|
|
|
|
|
7. |
|
|
|
|
|
15 |
|
|
|
|
|
|
|
|
3100 . . |
|
|
|
|
|
|
400 |
. ., |
|
10 |
|
|||
2600 . . |
|
|
|
|
|
|
500 . . |
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
5%- |
|
|
? |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
36 |
|
|
|
|
|
|
|
|
|
|
?
?
8. , () , , .
()
|
. |
|
|
|
5 |
. |
|
|
|
|
10,6 |
|
|
|
19,8 |
, |
|
|
, |
|
, 9,2 |
14,6 |
. |
0,05, |
|
|
|
|
|
. |
|
|
|
|
|
9. |
|
|
|
|
|
10 12 |
. |
|
|
|
135 |
136 |
|
4 |
6. |
|
|
0,01 |
|
|
. |
|
|
10. 9 , 13,7% 0,7%. 12 y 12,4% S x 1.1% . 0,05 ,
|
|
|
|
|
. |
|
|
|
11. |
|
|
|
|
|
|
|
|
|
|
|
|
: |
|
|
|
15 |
|
|
|
|
, |
20 |
. ./ |
|
|
|
|
|
5 . .; |
|
|
|
17 |
, |
|
|
|
25 |
. ./ |
|
|
|
|
4 . . |
|
, |
|
|
|
|
|
|
|
|
|
0,10. |
|
|
|
|
|
12. |
|
, |
|
|
|
|
|
|
|
|
. |
|
|
|
|
598 |
S |
= 74, |
|
572 |
S = 72. |
|
|
|
36. |
|
|
= 0,01. |
|
|
|
|
|
|
|
13. |
|
|
|
|
|
|
|
|
|
|
|
|
3 |
. |
|
|
|
, |
|
|
|
, |
: |
|
|
|
|
|
15,6; 18,2; 13,3. |
|
|
|
|
|
|
|
|
17,1; 16,3; 18,8. |
|
|
|
|
|
|
|
, |
, |
|
5%, |
, |
|
|
|
? |
|
|
|
|
|
|
|
|
14. |
|
|
|
, |
|
|
|
|
|
|
|
|
|
|
, |
|
|
( ) |
|
66,6 |
S = 6,4 |
n = 36, |
( |
) 62,3 |
S = 6,7 |
n |
= 17, |
n |
|
|
|
. |
|
|
|
|
|
|
|
37 |
|
|
|
|
15. |
|
|
|
|
|
( |
) |
|
|
|
|
. |
|
|
|
: |
|
|
|
|
|
( |
) |
. |
|
|
|
81,5 |
129,4 |
115,3 |
148,1 |
173,6 |
76,8 |
102,3 |
- |
|
79,1 |
105,3 |
181,2 |
120,3 |
143,4 |
165,8 153,8 |
178,3 |
0,01, |
|
, . . |
? |
16. |
, |
|
|
|
|
|
. |
: |
x |
1 |
70.2 |
n1 4 |
|
|
2 70.5 |
n2 = |
9. |
|
|
|
|
|
|
x |
|
|
|
|
|
|
||||
=0,05 |
|
, |
|
|
|
|
|
|
||
|
|
|
, |
|
|
|
|
|
|
|
17. |
S1 |
0.50 S2 |
10 . |
|
|
|
|
|
||
|
|
|
30 |
|
|
|
|
|||
|
|
|
|
|
. |
31 |
32 |
, |
||
|
|
|
|
40 |
60. |
0.95 |
|
|
|
|
|
|
|
|
|
|
. |
|
|
|
|
|
4.3. |
|
|
|
|
|
|
|
|
|
|
|
|
|
(F- |
|
|
) |
|
|
|
|
F- |
|
: X1,..,Xn |
Y1,..,Y |
|
|
|
|
|
|
(n, m < 100) |
|
|
|
|
|
|
2 |
2 |
|
|
|
y . |
|
|
|
|
x |
y |
|
|
|
|
|
|
|
|
|
H 0 : |
2 |
2 |
|
|
|
|
|
|
|
x |
y . |
|
|
|
|
|
|
|
|
|
H1 : |
2 |
2 |
|
, |
|
|
|
|
|
x |
y . |
|
|
|
|
||
|
|
|
( |
|
2 ), |
|
( |
2 ), |
|
|
|
|
|
|
1 |
|
|
2 |
|
|
|
|
|
|
1 : |
2 |
2 |
|
|
|
|
|
|
|
1 |
2 . |
|
|
|
|
|
|
: . |
|
|
|
|
|
|
|
|
|
F- |
|
: |
|
|
|
|
1. |
|
|
|
|
|
|
|
|
, |
2. |
|
|
|
0 |
1, |
|
|
|
. |
|
|
|
|
X1,..,Xn |
Y1,..,Y |
n |
m |
.
3. |
S x2 |
S y2 . |
( S x2 S y2 ) S12 , - S22 .
38
F-
S 2
F 1 ,
S22
1 n1 1 |
2 |
n2 1, |
|
1 - |
|
|
|
|
|
|
. |
|
||||
4. |
|
|
|
|
|
|
|
|
|
F- |
( |
|
|
|
5) |
|
|
|
|
|
|
F- |
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
n1 |
1 |
|
2 |
n2 1. |
|
|
|
|
|
|
||
|
|
H1 : |
2 |
|
2 |
|
|
F ( |
|
, |
1, |
2 ) |
|
|||
|
|
x |
|
y |
|
|
|
|
||||||||
|
|
|
|
|
2 |
|
||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1, |
2 . |
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
H1 : |
|
2 |
|
2 |
|
|
|
|
F ( , 1, 2 ) . |
|
||||
|
|
|
x |
|
y |
|
|
|
|
|
||||||
F |
F |
, |
|
|
|
|
|
|
|
|
|
|
|
|
. |
|
|
. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
. |
|
|
|
|
|
|
|
21 |
|
|
|
|
, |
|
|
|
|
|
|
|
|
|
|
|
0,072, |
|
|
|
|
0,068 |
16 |
|
. |
|
|
|
|
|
|
|
|
, |
|
|
|
|
|
|
|
|
, |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0,05. |
|
. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1. |
H0 : |
2 |
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
A |
|
B . |
|
|
|
|
|
|
|
|
|
|
||||
|
H : |
2 |
|
2 |
, |
. |
., |
|
|
|
|
|
, |
|||
|
|
1 |
|
|
A |
|
B |
|
|
, |
|
. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
: |
|
|
= 0,05. |
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
. |
|
2.-3. () (§2):
S 2 |
|
|
n |
2 . |
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
S |
|
|
|
|
|
|
|
|
|
||
|
|
n |
|
|
|
|
|
|
|
|
||||||
|
|
|
1 |
|
20 |
|
|
|
|
|
|
|
||||
nA = 21, S A2, |
|
|
0,072 |
0,0756 , |
|
|||||||||||
20 |
|
|
|
|||||||||||||
|
|
|
|
|
|
|
1 |
|
|
|
|
|||||
nB = 16, SB2 |
, |
|
|
16 |
|
|
0,068 |
|
0,0725 . |
|
||||||
|
16 |
1 |
|
|
||||||||||||
|
|
|
|
|
S 2 |
2 |
|
2 |
|
|
||||||
|
|
|
|
|
max(S |
, S |
) |
max(0,0756;0, 0725) 0, 0756 . |
||||||||
|
|
|
|
|
1 |
|
|
|
|
|
A, |
|
, |
|
|
|
|
|
|
|
|
|
|
|
|
F- |
|
|
|
|
|
||
|
S 2 |
0,0756 |
|
|
|
|
|
|
|
|
|
|||||
f |
1 |
|
|
|
|
|
|
1,043, |
|
|
|
|
||||
S22 |
0,0725 |
|
|
|
|
|||||||||||
|
|
|
|
|
|
|
|
|
|
|||||||
1 = n1 |
1 = 21 1 = 20 |
|
|
2 = n2 |
1 = 16 |
|
1 = 15. |
4. 5= 0,05;
F ( ; 2 ) F ( 0,05; 20;15)2,33.
39
: |
F F (1,043 < 2,33), |
0 |
. |
, |
0,05 |
|
|
|
, . . |
|
|
. , ,
.
§ 4.3.
,
, .
1.
( 40 ).
7,22 7,79 . . ,
0,10?
2.
10
.
: 0,17; 0,25. 0,10
.
3.
15 . ,
, , 0,25 0,49.
=0,1,
.
4. . 10 . , . 0,00064 (), 0,00039 (). = 0,01 ,
|
|
, |
|
|
. |
|
|
5. |
|
|
. |
12 |
, |
, 13 |
. |
|
|
|
: 0,76 |
( |
), 0,52 ( |
). |
0,10 |
|
, |
|
. |
6. . . ,
, .
.
40