Question

1£®ÎªÁ˽ⴺ¼¾Öçҹβî´óСÓëijÖÖ×Ó·¢Ñ¿¶àÉÙÖ®¼äµÄ¹Øϵ£¬·Ö±ð¼Ç¼ÁË4ÔÂ1ÈÕÖÁ4ÔÂ5ÈÕÿÌìµÄÖçҹβîÓëÿÌì100¿ÅÖÖ×Ó½þÅݺóµÄ·¢Ñ¿Êý£¬µÃµ½Èçϱí¸ñ£º

ÈÕÆÚ	4ÔÂ1ÈÕ	4ÔÂ2ÈÕ	4ÔÂ3ÈÕ	4ÔÂ4ÈÕ	4ÔÂ5ÈÕ
βîx¡ãC	12	11	13	10	8
·¢Ñ¿ÂÊy¿Å	26	25	30	23	16

£¨1£©´ÓÕâ5ÌìÖÐÈÎÑ¡2Ì죬ÇóÖÁÉÙÓÐÒ»ÌìÖÖ×Ó·¢Ñ¿Êý³¬¹ý25¿ÅµÄ¸ÅÂÊ£»
£¨2£©Çë¸ù¾Ý4ÔÂ1ÈÕ¡¢4ÔÂ2ÈÕ¡¢4ÔÂ3ÈÕÕâ3ÌìµÄÊý¾Ý£¬Çó³öy¹ØÓÚxµÄÏßÐԻع鷽³Ì$\widehat{y}$=$\widehat{b}$x+$\widehat{a}$£»
£¨3£©¸ù¾Ý£¨2£©ÖÐËùµÃµÄÏßÐԻع鷽³Ì£¬Ô¤²âβîΪ16¡ãCʱ£¬ÖÖ×Ó·¢Ñ¿µÄ¿ÅÊý£®
²Î¿¼¹«Ê½£º$\widehat{b}$=$\frac{\sum_{i=1}^{n}{x}_{i}{y}_{i}-n\overline{x}\overline{y}}{\sum_{i=1}^{n}{x}_{i}^{2}-n£¨\overline{x}£©^{2}}$£¬$\widehat{a}$=$\widehat{y}$-$\widehat{b}$x£®

Answer 1

·ÖÎö £¨1£©ÀûÓöÔÁ¢Ê¼þµÄ¸ÅÂʼÆËãËùÇóµÄ¸ÅÂÊÖµ£»
£¨2£©¼ÆËã$\overline{x}$¡¢$\overline{y}$£¬Çó³ö»Ø¹éϵÊý$\widehat{b}$£¬$\widehat{a}$£¬Ð´³ö»Ø¹é·½³Ì£»
£¨3£©ÀûÓûع鷽³Ì£¬¼ÆËãx=16ʱ$\widehat{y}$µÄÖµ¼´¿É£®

½â´ð ½â£º£¨1£©´ÓÕâ5ÌìÖÐÈÎÑ¡2Ì죬ÖÁÉÙÓÐÒ»ÌìÖÖ×Ó·¢Ñ¿Êý³¬¹ý25¿ÅµÄ¸ÅÂÊΪ
P=1-$\frac{{C}_{3}^{2}}{{C}_{5}^{2}}$=$\frac{7}{10}$£»
£¨2£©Çë¸ù¾Ý4ÔÂ1ÈÕ¡¢4ÔÂ2ÈÕ¡¢4ÔÂ3ÈÕÕâ3ÌìµÄÊý¾Ý£¬
¼ÆËã$\overline{x}$=$\frac{1}{3}$¡Á£¨12+11+13£©=12£¬
$\overline{y}$=$\frac{1}{3}$¡Á£¨26+25+30£©=27£¬
»Ø¹éϵÊýΪ$\widehat{b}$=$\frac{\sum_{i=1}^{n}{x}_{i}{y}_{i}-n\overline{x}\overline{y}}{\sum_{i=1}^{n}{x}_{i}^{2}-n£¨\overline{x}£©^{2}}$=$\frac{12¡Á26+11¡Á25+13¡Á30-3¡Á12¡Á27}{{12}^{2}{+11}^{2}{+13}^{2}-3{¡Á12}^{2}}$=$\frac{5}{2}$£¬
$\widehat{a}$=$\overline{y}$-$\widehat{b}$$\overline{x}$=27-$\frac{5}{2}$¡Á12=-3£¬
¡ày¹ØÓÚxµÄÏßÐԻع鷽³ÌΪ$\widehat{y}$=$\frac{5}{2}$x-3£»
£¨3£©¸ù¾Ý£¨2£©ÖÐËùµÃµÄÏßÐԻع鷽³Ì£¬¼ÆËãx=16ʱ£¬$\widehat{y}$=$\frac{5}{2}$¡Á16-3=37£»
¼´Ô¤²âβîΪ16¡ãCʱ£¬ÖÖ×Ó·¢Ñ¿µÄ¿ÅÊýΪ37£®

µãÆÀ ±¾Ì⿼²éÁËÏßÐԻعéÖ±Ïß·½³ÌµÄÇó·¨ÓëÓ¦ÓÃÎÊÌ⣬ÊÇ»ù´¡Ì⣮