Если не известны математическое ожидание и дисперсия, то доверительный интервал для математического ожидания m рассчитывается по следующей формуле
 .
Здесь  ,  – рассчитанные ранее точечные оценки,  – объем выборки,  – квантиль распределения Стьюдента, который определяется по таблицам,  – было рассчитано в предыдущем пункте.
При больших степенях свободы ( ) распределение Стьюдента стремится к нормальному закону, при этом условии квантиль можно смотреть по таблице нормального распределения.
 .
(Все это подставляем в формулу
 ,
 .
Таким образом, математическое ожидание с вероятностью 90% принадлежит интервалу  .
Доверительный интервал для дисперсии вычисляется по следующей формуле
 .
Значение  при больших степенях свободы приближенно можно вычислить по следующей формуле  .
Таким образом, в нашем случае
Следует заметить, что  , поэтому  , а квантиль  был найден ранее по таблице.
Подставляя полученные результаты в формулу для вычисления доверительного интервала, получим
 ,
 .
Следовательно, с вероятностью 90% дисперсия генеральной совокупности принадлежит интервалу  .
Данные к лабораторной работе
№ вар
|
Выборка
|
|
16,56; 15,76; 13,29; 12,09; 16,89; 12,46; 16,83; 15,38; 13,64; 15,46; 8,34; 11,34; 10,56; 16,78; 13,45; 20,30; 12,59; 14,59; 7,39; 5,87; 16,69; 9,32; 13,96; 9,71; 8,06; 14,79; 13,66; 14,48; 15,73; 19,38; 11,78; 14,45; 15,89; 13,62; 12,01; 13,84; 17,91; 15,60; 11,00; 8,08; 9,34; 3,95; 5,74; 17,21; 11,13; 10,47; 14,65; 6,99; 9,36; 7,82; 8,97; 14,08; 13,19; 10,61; 11,88; 4,56; 9,19; 12,35; 9,38; 11,59; 1,75; 5,33; 11,84; 14,56; 5,37; 12,48; 8,25; 10,46; 7,51; 12,33; 11,04; 12,97; 13,05; 6,58; 5,21; 2,15; 1,18; 8,68; 1,38; 8,03; 10,53; 7,47; 10,56; 12,39; 12,87; 11,32; 13,46; 16,51; 3,17; 13,19; 8,26; 6,72; 11,54; 5,40; 14,23; 12,12; 8,41; 10,86; 10,31; 12,90; 11,36; 12,35;
|
|
15,46; 7,38; 13,67; 12,92; 7,37; 17,17; 19,01; 25,38; 12,42; 10,06; 21,23; 18,15; 21,85; 15,48; 16,45;20,74; 13,94; 20,38; 18,47; 21,65; 17,47; 20,58; 18,54; 17,60; 20,38; 21,32; 18,16; 14,37; 19,22; 4,19; 19,17; 13,69; 18,79; 20,94; 24,28; 16,57; 20,07; 19,53; 17,44; 22,43; 18,36; 20,75; 18,33; 16,43; 14,34; 13,66; 5,54; 22,57; 8,04; 18,96; 15,23; 19,38; 14,59; 11,95; 12,06; 10,55; 16,90; 21,40; 17,99; 12,44; 14,00; 10,03; 14,67; 17,23; 10,08; 9,85; 10,77; 1,33; 1,86; 12,68; 2,73; 15,17; 4,66; 9,89; 15,33; 18,74; 16,44; 9,78; 12,36; 14,50; 11,66; 14,87; 16,38; 16,890; 15,46; 17,36; 20,38; 7,97; 13,34; 12,00; 10,36; 14,67; 15,78; 8,77; 14,45; 15,67; 13,56; 16,06; 13,25;
|
|
14,57; 13,68; 10,77; 11,47; 4,57; 9,67; 12,67; 9,58; 11,29; 1,01; 5,36; 11,34; 14,45; 5,00; 12,10; 8,56; 10,20; 7,46; 12,30; 11,56; 12,57; 13,03; 6,40; 5,67; 2,85; 1,50; 8,34; 1,60; 8,78; 10,57; 7,66; 14,40; 12,46; 12,67; 11,66; 13,67; 16,94; 3,76; 20,57; 18,56; 16,59; 17,45; 20,00; 15,46; 19,19; 18,70; 17,35; 19,26; 12,86; 15,65; 14,13; 19,09; 16,07; 22,00; 14,61, 17,36; 9,23; 8,34; 19,16; 17,78; 12,79; 13,58; 18,88; 17,55; 11,90; 16,73; 18,74; 23,80; 15,33; 19,84; 17,27; 15,96; 16,95; 21,84; 11,46; 12,47; 10,86; 18,85; 14,65; 15,44; 20,80; 7,46; 4,81; 16,67; 18,24; 15,67; 13,83; 12,18; 14,13; 13,78; 8,35; 6,27; 11,97; 5,31; 14,15; 12,41; 8,85; 10,79; 10,11; 12,84; 11,14; 12,07;
|
|
17,46; 19,45; 25,78; 12,45; 10,67; 21,45; 18,54; 21,55; 15,48; 16,45; 20,48; 13,55; 20,55; 18,15; 21,05; 17,48; 20,19; 18,54; 17,54; 20,54; 21,21; 14,85; 14,51; 19,68; 14,46; 19,46; 13,31; 18,88; 20,24; 18,46; 11,74; 17,25; 16,48; 24,44; 17,44; 20,74; 14,74; 15,74; 12,88; 19,68; 17,68; 19,46; 20,32; 16,25; 20,60; 17,00; 16,68; 19,74; 20,65; 11,65; 9,65; 17,87; 13,74; 20,54; 2,55; 18,54; 3,45; 18,54; 12,25; 17,64; 19,84; 23,15; 15,64; 19,46; 16,45; 17,25; 17,54; 21,55; 16,55; 19,54; 15,14; 13,01; 12,01; 2,54; 4,44; 7,05; 15,54; 21,45; 11,21; 13,41; 9;01; 24,26; 16,16; 20,46; 19,62; 17,24; 22,54; 18,54; 20,54; 18,45; 16,44; 14,58; 13,55; 5,55; 22,02; 8,51; 18,51; 15,56; 19,21; 14,51; 11,45; 12,51; 10,51;
|
|
14,25; 16,41; 9,54; 18,21; 15,25; 21,54; 16,54; 16,51; 14,51; 15,21; 19,64; 15,34; 7,64; 22,15; 15,54; 13,50; 11,54; 16,24; 10,54; 7,51; 22,15; 15,56; 13,54; 12,54; 13,54; 10,45; 17,06; 15,34; 17,64; 18,51; 14,54; 15,54; 18,64; 14,54; 17,54; 17,55; 13,09; 11,65; 10,65; 4,96; 2,65; 5,21; 19,21; 13,54; 12,54; 16,54; 8,85; 11,96; 9,48; 13,74; 17,98; 12,64; 13,54; 10,51; 17,50; 15,00; 17,54; 18,54; 14,41; 15,54; 18,95; 14,47; 17,47; 18,30; 15,38; 11,54; 16,61; 11, 63; 16,61; 10,65; 15,05; 17,65; 21,64; 13,45; 17,68; 11,45; 16,67; 10,61; 15,64; 17,61; 21,65; 13,61; 17,56; 16,54; 14,44; 15,05; 19,16; 15,26; 17,59; 13,56; 11,06; 10,44; 2,96; 5,54; 19,58; 13,54; 12,54; 16,59; 8,54; 11,26; 9,18; 13,26; 18,54;
|
|
18,65; 11,87; 17,47; 16,54; 24,65; 17,24; 20,21; 14,51; 15,81; 12,94; 19,64; 17,65; 20,54; 16,41; 20,55; 17,44; 16,54; 19,54; 20,85; 11,55; 9,51; 17,44; 13,51; 20,51; 2,10; 18,00; 3,51; 13,05; 16,64; 9,64; 7,69; 22,51; 15,64; 13,65; 12,45; 13,65; 10,55; 17,69; 15,88; 16,68; 18,54; 14,00; 15,05; 18,06; 14,69; 17,57; 18,51; 14,54; 11,05; 17,54; 10,54; 16,54; 11, 47; 15,75; 17,54; 21,65; 13,24; 17,21; 16,05; 14,25; 15,54; 19,09; 15,04; 17,61; 13,47; 11,54; 10,54; 2,54; 5,15; 19,48; 13,45; 12,54; 16,54; 8,54; 11,14; 9,50; 13,57; 17,84; 18,47; 12,06; 17,55; 19,05; 23,02; 15,47; 19,17; 16,45; 17,64; 17,57; 21,67; 16,55; 19,58; 15,68; 13,58; 12,64; 4,26; 7,39; 15,56; 21,69; 11,07; 13,04; 9,40;
|
|
20,54; 18,54; 16,45; 17,78; 20,48; 15,58; 19,58; 18,48; 17,42; 19,12; 12,25; 15,45; 14,55; 19,03; 16,51; 22,45; 14,54; 17,05; 9,45; 8,54; 19,54; 17,55; 12,25; 13,65; 18,56; 17,54; 11,54; 16,06; 18,64; 23,64; 15,44; 19,54; 14,54; 16,05; 9,54; 7,54; 22,06; 15,64; 13,54; 12,65; 13,54; 10,44; 17,05; 15,96; 17,54; 18,54; 14,54; 15,12; 18,14; 14,47; 17,54; 18,44; 15,25; 11,55; 16,24; 11,54; 16,41; 10,15; 15,06; 17,54; 21,15; 13,54; 17,26; 16,68; 14,45; 15,54; 19,15; 15,24; 17,64; 13,24; 11,24; 10,24; 2,34; 5,15; 19,25; 13,03; 12,06; 16,45; 8,45; 11,54; 9,54; 13,45; 18,54; 17,06; 15,42; 16,16; 21,26; 11,25; 12,54; 10,64; 18,4; 14,56; 15,46; 20,65; 7,45; 4,54; 16,54; 18,64; 15,64; 15,43; 12,54; 14,56;
|
|
15,04; 17,25; 24,54; 11,57; 16,15; 10,28; 15,55; 17,34; 21,68; 13,58; 17,67; 17,65; 19,64; 14,54; 11,44; 9,00; 20,64; 19,64; 11,54; 16,54; 10,49; 7,34; 22,57; 15,58; 13,54; 12,25; 13,54; 10,44; 17,25; 15,57; 17,54; 18,54; 14,14; 15,54; 18,54; 14,46; 17,41; 15,48; 15,48; 21,51; 14,58; 16,54; 14,44; 15,13; 19,05; 15,51; 17,24; 13,15; 11,12; 10,15; 2,01; 5,21; 19,15; 13,51; 12,81; 16,17; 8,12; 11,78; 9,45; 13,54; 17;15; 15,44; 11,36; 20,22; 15,44; 19,45; 17,41; 16,41; 9,71; 11,44; 10,51; 8,44; 19,22; 16,00; 13,05; 12,10; 17,41; 17,58; 10,14; 15,23; 17,44; 21,74; 14,46; 17,41; 18,18; 15,08; 16,23; 21,27; 16,14; 19,54; 12,95; 15,81; 12,41; 11,67; 4,65; 5,78; 19,06; 14,43; 13,85; 18,65; 10,14; 12,13;
|
|
15,77; 16,19; 11,84; 9,81; 16,46; 10,42; 15,55; 17,16; 21,51; 11,45; 10,37; 14,05; 10,59; 11,77; 17,61; 5,15; 2,19; 17,52; 13,19; 16,15; 11,66; 18,45; 12,44; 13,17; 19,51; 23,99; 15,51; 19,62; 18,44; 14,37; 17,51; 16,51; 18,64; 14,16; 16,02; 11,06; 14,22; 16,54; 9,41; 7,66; 22,51; 15,81; 13,51; 12,05; 13,13; 10,21; 17,24; 15,19; 17,05; 18,57; 14,11; 15,18; 18,55; 14,27; 17,19; 18,07; 15,03; 11,15; 16,45; 11,34; 13,61; 17,91; 16,65; 14,48; 15,64; 19,84; 15,55; 17,64; 13,57; 11,19; 10,99; 2,45; 5,37; 19,35; 13,06; 12,04; 16,22; 8,94; 11,51; 9,97; 13,24; 18,64; 9,51; 8, 81;16,06; 15,64; 12,51; 15,54; 20,21; 14,18; 18,15; 15,51; 17,16; 8,55; 14,51; 16,20; 9,07; 7,22; 20,64;13,51; 16,94; 9,49;
|
|
17,88; 11,90; 16,30; 11,55; 15,84; 17,66; 14,81; 17,09; 6,59; 16,64; 20,84; 14,08; 13,51; 16,34; 9,58; 7,49; 22,81; 15,88; 13,47; 12,95; 13,45; 10,78; 17,79; 15,05; 16,64; 18,52; 14,88; 15,15; 18,31; 14,51; 17,06; 18,11; 14,91; 11,95; 21,44; 13,22; 17,81; 16,97; 14,64; 15,51; 19,18; 15,34; 17,82; 13,83; 11,93; 10,51; 2,22; 5,84; 19,64; 13,44; 12,81; 16,19; 8,24; 11,55; 9,51; 13,71; 17,77; 17,83; 11,66; 16,80; 21,64; 9,82; 16,64; 12,09; 8,51; 1,34; 12,86; 18,51; 14,34; 8,55; 16,91; 10,73; 15,75; 17,95; 16,81; 8,31; 24,64; 17,67; 20,02; 2,64; 18,07; 4,22 4,84; 7,24; 15,37; 19,99; 13,51; 18,59; 20,58; 7,53; 11,56; 19,48; 13,07; 12,16; 17,16; 19,18; 20,74; 7,75; 17,06; 2,07; 10,64; 12,01;
|
|
11,48; 16,15; 10,88; 7,72; 22,08; 15,16; 13,19; 12,14; 13,82; 10,37; 17,44; 15,84; 17,66; 18,81; 14,06; 15,08; 18,16; 14,04; 17,64; 18,37; 15,91; 21,31; 16,81; 11,11; 16,85; 10,54; 15,64; 17,94; 21,07; 13,16; 17,17; 16,19; 14,37; 15,08; 19,09; 15,37; 17,38; 13,92; 11,66; 10,82; 2,07; 5,06; 19,03; 13,34; 12,45; 16,46; 8,56; 11,59; 9,54; 13,06; 17,15; 14,57; 16,37; 9,38; 7,91; 22,27; 15,16; 13,19; 12,16; 13,15; 10,29; 17,36; 15,38; 17,64; 18,94; 14,16; 15,13; 18,18; 14,24; 17,22; 18,29; 15,54; 11,88; 16,12; 11,06; 16,56; 10,51; 15,61; 17,67; 21,94; 13,34; 17,51 16,81; 14,87; 15,35; 19,65; 15,44; 17,68; 13,46; 11,05; 10,07; 2,21; 5,64; 19,65; 13,91; 12,38; 16,21; 8,15; 11,33; 9,51; 13,32; 18,98;
|
|
5,04; 7,16; 13,55; 12,31; 7,09; 25,67; 18,16; 16,19; 17,27; 20,06; 25,13; 19,19; 18,46; 17,16; 19,46; 12,19; 15,18; 14,37; 19,38; 16,38; 22,26; 14,29; 17,44; 19,01; 8,06; 19,12; 17,00; 12,16; 13,32; 18,73; 17,72; 21,08; 16,03; 18,13; 23,73; 15,46; 19,45; 17,15; 15,46; 18,48; 21,49; 11,45; 22,45; 10,25; 18,28; 14,35; 15,46; 20,56; 17,65; 14,94; 16,99; 18,91; 15,85;13,16; 12,46; 14;76; 16,11; 21,46; 17,76; 12,73; 14,53; 10,75; 14,05; 17,09; 10,45; 9,37; 10,44; 1,16; 1,45; 12,48; 2,77; 15,73; 4,49; 9,49; 15,48; 8,55; 16,08; 9,67; 12,62; 14,92; 11,68; 14,45; 6,49; 16,08; 15,45; 17,75; 12,07; 7,18; 13,19; 12,67; 10,37; 14,38; 15,66; 8,04; 14,37; 15,56; 13,67; 16,88; 13,59;
|
|
16,54; 19,81; 25,31; 15,09; 21,46; 16,08; 11,88; 16,62; 10,91; 15,81; 17,73; 21,51; 13,55; 12,87; 10,56; 21,77; 18,64; 21,65; 15,84; 16,67; 20,84; 13,84; 20,67; 18,87; 23,99; 17,82; 20,11; 18,08; 17,07; 20,64; 20,84; 18,54; 14,87; 19,66; 14,18; 19,78; 13,78; 18,12; 20,19; 24,18; 16,84; 11,18; 16,55; 10,87; 7,83; 22,35; 15,29; 13,28; 12,44; 13,16; 10,33; 17,09; 15,48; 17,28; 18,27; 14,24; 15,24; 18,46; 14,45; 17,61; 18,76; 17,79; 16,02; 14,11; 15,30; 19,55; 15,63; 17,13; 13,75; 11,04; 10,93; 2,45; 5,47; 19,76; 13,68; 12,69; 16,72; 8,08; 11,61; 9,75; 13,73; 17,09; 20,22; 19,45; 17,18; 22,79; 18,08; 20,27; 18,37; 16,66; 14,87; 13,05; 5,14; 24,16; 18,76; 18,84; 15,85; 19,67; 14,30; 11,19; 12,47; 10,01;
|
|
13,44; 16,16; 9,48; 7,02; 22,57; 15,84; 13,48; 12,67; 13,08; 11,23; 16,13; 10,33; 7,28; 22,39; 15,34; 13,58; 12,22; 13,64; 10,19; 17,18; 15,45; 17,07; 18,08; 14,12; 15,51; 18,84; 14,31; 17,03; 18,21; 15,61; 21,74; 16,18; 11,47; 16,11; 10,07; 15,32; 17,37; 21,64; 13,51 ;17,12; 16,16; 14,45; 15,45; 19,85; 15,86; 17,19; 13,54; 11,04; 10,98; 2,99 ; 5,51; 19,15; 13,42; 12,72; 16,73; 8,28; 11,17; 9,44; 13,16; 17,44; 10,26; 17,97; 15,16; 16,19; 18,46; 14,54; 15,68; 18,06; 14,62; 17,18; 18,49; 14,45; 11,47; 17,49; 11,46; 16,57; 11,55; 15,19; 17,45; 21,66; 13,21; 17,81; 16,07; 14,34; 15,61; 19,51; 15,41; 17,52; 13,82; 11,72; 10,83; 2,44; 5,56; 19,69; 13,58; 12,81; 16,42; 8,72; 11,12; 9,48; 13,45; 17,89;
|
|
5,14; 7,75; 13,16; 12,09; 7,45; 16,75; 21,85; 17,96; 14,92; 17,80; 6,02; 16,45; 20,57; 14,66; 17,19; 11,45; 16,48; 21,34; 9,15; 16,49; 12,78; 8,15; 1,44; 12,44; 18,19; 14,45; 8,60;16,15; 10,41; 15,63; 17,33; 16,06; 8,15; 24,48; 17,87; 20,81; 2,45; 18,55; 4,54; 4,49; 7,74; 15,45 ; 19,42; 13,43; 18,15; 20,44; 7,51; 11,44; 19,16; 13,76; 12,78; 17,79; 19,08; 20,51; 7,92; 17,97; 2,93; 10,92; 12,84; 12,16; 14,44; 10,51; 14,91; 17,83; 10,77; 9,15; 10,54; 4,47; 11,61; 12,57; 2,49; 15,18; 4,18; 9,19; 15,77; 18,44; 16,46; 9,18; 12,44; 14,62; 11,48; 14,77; 16,44; 16,12; 15,44; 17,45; 20,81; 7,44; 13,15; 12,75; 10, 19;14,47; 15,66; 8,49; 14,48; 15,46; 13,04; 16, 95; 14,08;
|
|
10, 51; 16,19; 5,74; 7,61; 13,84; 12,75; 7,46; 17,84; 12,49; 14,11; 9,72; 10,61; 4,45; 11,16; 12,55; 2,81; 15,46; 4,11; 9,14; 15,72; 18,55; 16,14; 9,16; 12,62; 14,96; 11,63;14,33; 16,09; 16,08; 15,45; 17,08; 20,07; 7,31; 13,64; 12,91; 10,95; 14,58; 15,54; 8,56; 14,24; 15,34; 13,16; 16,15; 14,45; 10,54; 17,84; 22,74; 15,24; 13,61; 12,17; 16,12; 21,22; 13,51; 10,35; 17,16; 15,77; 17,02; 18,16; 14,55; 15,46; 18,51; 14,48; 17,77; 19,16; 15,27; 10,46; 14,57; 17,22; 10,34; 21,35; 16,15; 11,75; 16,18; 10,78;15,66; 17,68; 21,95; 13,92; 16,54; 16,05; 14,22; 15,46; 19,03; 15,32; 17,62; 13,64; 11,67; 10,91; 2,22; 5,15; 19,55; 13,11 ;12,57; 16,67; 8,94; 11,52; 9,44; 13,25; 11,86;
|
|
11,34; 18,12;63; 10,73; 7,70; 1,45; 8,93; 1,76; 22,02; 15,67; 13,94; 8,88; 12,55; 13,74; 10,01; 17,95; 15,87; 17,97; 18,36; 14,29; 15,85; 18,86; 14,99; 17,56; 18,55; 15,83; 24,09; 16,87; 11,55; 16,78; 10,46; 15,66; 17,73; 21,66; 14,44; 13,72; 10,67;11,74; 4,85; 9,88; 12,56; 9,66; 11,39; 1,77; 5,73; 11,90; 14,05; 5,67; 12,48; 8,54; 10,71; 7,91; 12,13; 11,24; 12,49; 13,55; 6,38; 5,45; 2,54; 10,55; 7,88; 10,89; 12,48; 12,95; 11,83; 13,15; 16,28; 3,55; 13,62; 8,05; 6,06; 11,83; 5,54; 14,81; 12,80; 8,56; 10,52; 10,89; 12,73; 11,10; 12,12; 13,36; 17,66; 16,55; 14,77; 15,92; 19,66; 15,89; 17,49; 13,26; 11,28; 10,44; 2,47; 5,87; 19,11; 13,38; 12,79; 16,22; 8,35; 1,56; 23,44; 13,52; 16,99;
|
|
13,12; 16,45; 9,77; 7,71; 22,65; 15,83; 13,82; 12,99; 13,02; 10,97; 17,53; 15,77; 16,71; 18,06; 14,73; 15,83; 18,86; 14,77; 17,39; 18,19; 14,16; 11,30; 17,33; 11,45l; 16,66; 11,66; 15,83; 17,01; 14,72; 16,55; 9,28; 7,16; 22,22; 15,56; 13,32; 12,88; 13,60; 10,43; 17,55; 15,62; 17,88; 18,63; 14,84; 15,44; 18,83; 14,67; 17,17; 18,48; 15,84; 11,88; 16,83; 11,04; 16,84; 10,76; 15,77; 17,45; 21,23; 13,49; 17,96; 16,78; 14,39; 15,94; 19,84; 15,92; 17,59; 13,77; 11,94; 10,15; 2,85; 5,99; 19,01; 13,03; 12,89; 16,33; 8,78; 11,80; 9,97; 13,04; 18,00; 21,06; 13,83; 17,41; 16,27; 14,23; 15,34; 19,62; 15,85; 17,33; 13,75; 11,83; 10,01; 2,74; 5,22; 19,56; 13,94; 12,93; 16,02; 8,76; 11,82; 9,67; 13,48; 17,65;
|
|
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