Āน่üยการเรียนรู้ที่ 2 การแจกแจงคüามน่าจะเป็น 50 การแจกแจงปกติ(Normal Distribution) ในทางÿถิติข้อมูลที่เก็บมาจำนüนมาก ÿ่üนใĀญ่จะÿมมติใĀ้มีการแจกแจงปกติและมีพารามิเตอร์ที่ ÿำคัญอยู่ 2 ค่า คือ ค่าเฉลี่ยเลขคณิต( ) และคüามแปรปรüน( ) 2 ตัüแปรÿุ่ม x = − ( , ) เป็นตัüแปรÿุ่มที่มีการแจกแจงปกติ เขียนแทนด้üยÿัญลักþณ์ ∼ (, 2) ถ้าฟังก์ชันคüามĀนาแน่นของคüามน่าจะเป็นของตัüแปรÿุ่ม x คือ ( ) 2 1 2 1 2 x ; , 2 x f e − − = ซึ่งกราฟของฟังก์ชันคüามĀนาแน่นของคüามน่าจะเป็นของการแจกแจงปกติเป็นเÿ้นโค้งรูประฆังคü่ำ (bell-shaped curve) Āรือเÿ้นโค้งปกติ (normal curve) การแจกแจงแบบปกติที่มีค่าเฉลี่ยเลขคณิต( ) และÿ่üนเบี่ยงเบนมาตารฐาน( ) ถ้าตัüแปรÿุ่ม x = − ( , ) มีการแจกแจงปกติ ( ) 2 N , แล้üค่าคาดĀมายและคüามแปรปรüน ของตัüแปรÿุ่ม X คือ E x( ) = และ ( ) 2 V x = เนื่องจาก ∼ (, 2) เป็นตัüแปรÿุ่มชนิดต่อเนื่อง การĀา Pa x b ( ) ทำได้โดยĀาพื้นที่ใต้เÿ้นโค้งปกติ ระĀü่าง x a = ถึง x b = นั่นคือ ( ) 2 1 1 2 2 b x a Pa x b e − − = เพื่อคüามÿะดüกจึงทำการปรับตัüแปรÿุ่ม ∼ (, 2) ที่มีการแจกแจงปกติใĀ้อยู่ในรูป ตัüแปรÿุ่ม ∼ (0,1) ซึ่งมีการแจกแจงปกติมาตรฐาน ด้üยคüามÿัมพันธ์ x Z − = ถ้าใĀ้ 1 a Z − = และ 2 b Z − = จะได้ü่า
Āน่üยการเรียนรู้ที่ 2 การแจกแจงคüามน่าจะเป็น 51 ( ) ( ) 2 2 1 1 2 1 2 1 2 z z z axb Pa x b P Pz Z z e dz − −−− = = = เราเรียกเÿ้นโค้งปกติจากฟังก์ชัน 1 2 2 1 ( ) 2 z fz e − = ü่า เÿ้นโค้งปกติมาตรฐาน โดยที่พื้นที่ใต้เÿ้นโค้ง ÿามารถĀาได้จากตารางที่ 2 ของภาคผนüก เนื่องจากตัüเลขในตารางแทน P Zz (− ) ดังนั้น การĀา ( ) Pz Z z 1 2 ทำได้โดย ( ) ( ) P Zz P Zz − − − 2 1 พิจารณา ( ) P Zz − 2 ดังรูป พิจารณา ( ) P Zz − 1 ดังรูป จะได้ü่า ( ) ( ) P Zz P Zz − − − 2 1 เท่ากับ ( ) Pz Z z 1 2 ดังรูป
Āน่üยการเรียนรู้ที่ 2 การแจกแจงคüามน่าจะเป็น 52 ตัüอย่าง 1 กำĀนด Z เป็นตัüแปรÿุ่มที่มีการแจกแจงปกติมาตรฐาน ใĀ้Āา 1) P Z( 2.4) 2) P Z( 1.52) ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 3) P Z (1.2 2.8 ) 4) P Z (− 2.23 3.2) ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ตัüอย่าง 2 กำĀนด x เป็นตัüแปรÿุ่มที่มีการแจกแจงปกติ ซึ่งมีค่าเฉลี่ยเท่ากับ 18 และÿ่üนเบี่ยงเบนมาตรฐานเท่ากับ 2.5 ใĀ้Āา 1) P x( 12) 2) P x( 25) ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 3) P x (16 20 ) 4) P x (20 24 ) ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. in' 0 2.4 D 1.52 6999077878 P(2<2.41 = D. 9918 Plz < 1.52 ) = D. 9357 PIZ > 1.521 = 2 - 0.9357 = 0.0643 Pl1o2< 2<281 = 0.9974 - 0.8849 - 2.23 b 3.2 = D. 1125 ÷ PIZC -2231=0.0129 P(2<327--0.9993 Pc-2222<3.21--0.9993 - 0.0129 = 0.9864 # D. 0026 & Plz> a. s ) 12 - 18 25-28 PCZC 25 1 28 P(2<-241 Plz > 281 = D. 0022 6Th 757J Plz < 2.81 = Do 9974 PIZ > 281 = 1 - D. 9974 F) = 0.0026 # i -24 O P 12<-2.8) = 0.0026 # "# to p( 16-18 p(23%1<2 < 24-18 2. s < 2 < % ) 2.s ) Pt-0.8<2 < D. B) Plo.BE 22241 =D. 7881 - D. 2119 = D. 9913 < 0.7881 = 0.5762 # = 0.2037 #
Āน่üยการเรียนรู้ที่ 2 การแจกแจงคüามน่าจะเป็น 53 ตัüอย่าง 3 ใĀ้Āาค่า a ที่ทำใĀ้ 1) PZ a ( =) 0.0274 2) PZ a ( =) 0.0116 ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ตัüอย่าง 4 คะแนนÿอบüิชาÿถิติของนักเรียนมีการแจกแจงแบบปกติ โดยมีคะแนนเฉลี่ย 75 คะแนน และÿ่üนเบี่ยงเบนมาตรฐาน 12 คะแนน ใĀ้Āา 1) คüามน่าจะเป็นที่นักเรียนจะÿอบได้คะแนนมากกü่า 60 คะแนน ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 2) มีนักเรียนกี่เปอร์เซ็นต์ที่ÿอบได้คะแนนระĀü่าง 54 ถึง 90 คะแนน ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 3) ถ้าครูผู้ÿอนใĀ้เกรด 4 กับนักเรียนที่ÿอบได้คะแนนÿูงÿุด 15%นักเรียนจะต้องทำคะแนนได้เท่าไรจึงจะได้ เกรด 4 ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. D. 0274 q "" " • Tiki. Plz c- 1.921 = 0.0274 0 01=2.27 01 = - 2,92 1 - D. 0116 = D. 9884 Plz < 2.271 = 0.9884 Plz C- 2271=0.0226 D7ÑDNEF it = 756628 5=12 -1.25 0 PIX > 601 = Plz > 60-75 12 Ñ@ PIZC-1.251=0.7056 = Plz > -1.251 Ñ@ Plz > -1.251=0.8944 = 1 - P 12<-1.251 Plz > -1.251=0122%251 = 1- 0.1056 = 0.8944 # ~ = 0.8944# Pl54 (✗ < 90) -1.75 0 7.25 =P / 54-75<2 < 90-75 6Ñ@ ④7878 12 12 P / 2<1.25 ) = D. 8944 = Pl - 1.75 < 2<7.251 PCZC-1.751--0.0401 = 0.8944 - 0.0401 = 0.8543 # Ñ"="'s 1 6%017879 Pl -2<01 ) = 1- D. 15 = D. 85 P( 2<1.041 = D. 8508 1. 04 = ✗ - 75 ✗ = 12.48 + 75 22 ✗ = 87=48 1. 041221 = X - 75 E. Noggins 87 nI66WWÑÑfU
Āน่üยการเรียนรู้ที่ 2 การแจกแจงคüามน่าจะเป็น 54 ตัüอย่าง 5 คะแนนÿอบüิชาคณิตýาÿตร์ของนักเรียนชั้นมัธยมýึกþาปีที่ 6 Ā้องĀนึ่งมีการแจกแจงปกติ โดยมีค่าเฉลี่ยเลขคณิตและÿ่üนเบี่ยงเบนมาตรฐานเท่ากับ 72 และ 12 คะแนนตามลำดับ ถ้ากำĀนดใĀ้ P Z( − = 0.6745 0.25 ) และ P Z( = 1.2816 0.90 ) จงĀาเปอร์เซ็นต์ไทล์ที่ 25 และ เปอร์เซ็นต์ไทล์ที่ 90 ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 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................................................................................................................................................................................. YEONG U = 72 5=72 MI Pas = ? 662:Pgo= ? annoying z= Xgll > 0.25 n0.6745 µ = 72 66NWÑ7 - D. 6745 = ✗ - 72 72 - D.6745172)= ✗ - 72 - B. 0941-72 = ✗ 63.906 = ✗ ④ON Pzs ~ 64 # MPs, ~ = ? D)nsgtsmoz = ✗ - µ 6 66min17 1.2326 = ✗ -72 12 2. 28261121 = ☒ - 72 15.37921-72 = ✗ 8703792 = ✗ 5. Pro < 87 F-
Āน่üยการเรียนรู้ที่ 2 การแจกแจงคüามน่าจะเป็น 55 แบบฝึกĀัด 7 1. ใĀ้Āาคüามน่าจะเป็นของการแจกแจงปกติในแต่ละข้อต่อไปนี้ จากพื้นที่ใต้เÿ้นโค้งปกติมาตรฐานที่กำĀนดใĀ้ 1) 2) ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 3) 4) ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 2. กำĀนด Z เป็นตัüแปรÿุ่มที่มีการแจกแจงปกติมาตรฐาน ใĀ้Āา 1) P (Z > 0.16) 2) P (Z < -1.14) 3) P (-1.3 <Z<2.5) ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. Pl - 2.79<2 C- 7.98) + Plo < 2<7.571 PC -2.1342<-1.35)tP(1.82<242.24) = (0.0239-0.0743) 1- (0.9478-0.5000) = (0.0885-0.0166)+(0.9875-0.9656) = 0.4514 = D. 0938 Pc-2.7 < 2<01 PCZC - 2. 2) 1- PC-1.522<7-7)tPcocz< 1.g) + P 17.9<2 ( 2.81 = 0.5000 - 0.0035 = 0.01391-(0.7357-0.0668) + (0.9332-0-5000) + = D.4965 = D. 5421 (0.9974-0.9713) PIZ C- 0.161 = D. 1277 = D. 9938 - 0.0968 = 0.4364 = 0.8970
Āน่üยการเรียนรู้ที่ 2 การแจกแจงคüามน่าจะเป็น 56 4) P (1.56 <Z<2.92) 5) P (-2.63<Z<-0.1) 6) P (Z < 1.45) ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 3. กำĀนด X เป็นตัüแปรÿุ่มที่มีการแจกแจงปกติ ซึ่งมีค่าเฉลี่ยเลขคณิตเท่ากับ 150 และ ÿ่üนเบี่ยงเบนมาตรฐานเท่ากับ 25 ใĀ้Āา 1) P(X < 76) 2) P(X > 120) 3) P(X > 192) ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 4) P(180 <X <200) 5) P(68 < X < 100) 6) P(X > 220) + P(100 < X < 190) ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. =D-9982 - 0.9406 = D.4602-0.0043 = 0.9265 = 0.0576 = 0.4559 Plz < 76-7501 PIZ > 720-1501 PCZ > 792-1501 25 25 25 PCZC -2.967 = 0.0075 PIZ> - 7.21 PLZ > 1.681 PIZC 7.21 PCZC -7.68 ) =D. 8849 = 0.0465 P / 1.2<2 < 2) Pl - 3.28 < 2<-21 Plz > 2. 8) t Pl - 2C 2- < 1.61 = D.9772 - D. 8849 = 0.0228 = D. 0005 = 0-0026 1- (0.9452-0.0228) = 0.0923 = D. 0223 = 0.9250
Āน่üยการเรียนรู้ที่ 2 การแจกแจงคüามน่าจะเป็น 57 4. กำĀนด Z เป็นตัüแปรÿุ่มที่มีการแจกแจงปกติมาตรฐาน ใĀ้Āาค่า a ที่ทำใĀ้ 1) P (Z < a) = 0.0606 2) P (Z > a) = 0.3745 ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 3) P (-a <Z< a) = 0.9844 4) P (Z < a) = 0.5111 ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 5. คะแนนÿอบüิชาภาþาอังกฤþมีการแจกแจงปกติ มีค่าเฉลี่ยเลขคณิตเท่ากับ 70 คะแนนและÿ่üนเบี่ยงเบน มาตรฐานเท่ากับ 10 คะแนน นายนพมีคะแนนÿอบตรงกับตำแĀน่งเดไซล์ที่ 3.3 นายนพÿอบüิชาภาþาอังกฤþ ได้คะแนนเท่าไร ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 6. แผ่นเĀล็กที่ผลิตจากโรงงานแĀ่งĀนึ่งมีการแจกแจงปกติ มีคüามĀนาของแผ่นเĀล็กโดยเฉลี่ย 2 4.41 10− เซนติเมตร มีคüามแปรปรüนเป็น 6 4.41 10− เซนติเมตร2 ใĀ้Āาคüามน่าจะเป็นที่แผ่นเĀล็กคüามĀนาอยู่ ระĀü่าง 0.042 เซนติเมตร และ 0.0483 เซนติเมตร ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 01 = -1.55 - 01 = - D. 32 01 = 0.32 Plz ca) - Plz < -011=0.9844 a ÑÑ7 @ Effendis D. 02 - 0.03 Plzcoll 1- Plz c- a) = 1. DODD 20.03 2012<011--7.9844 01 = 2.42 33 -1. → Plz Lol ) =D.33 ✗ = 65.6 = 66 A:i6WW 01 = -0.44 ✗ - µ = - 0.44 o ✗ - 70 = - D. 44 10 Pco -042 ( ✗ < D- 04831 P 10.042 - 0.047 C Z C D. 0483 - D. 0491 ) D. 0022 D. 0027 PI - IL 2<21 = D. 9772 - 0.7587 = 0.8285 #
Āน่üยการเรียนรู้ที่ 2 การแจกแจงคüามน่าจะเป็น 58 7. น้ำĀนัก(กิโลกรัม)ของนักเรียนĀ้องĀนึ่งมีการแจกแจงปกติมาตรฐาน โดยมีน้ำĀนักเฉลี่ยเท่ากับ 50 กิโลกรัม และÿ่üนเบี่ยงเบนมาตรฐานเท่ากับ 4 กิโลกรัม ใĀ้Āา 1) คüามน่าจะเป็นที่นักเรียนมีน้ำĀนักมากกü่า 48 กิโลกรัม ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 2) คüามน่าจะเป็นที่น้ำĀนักของนักเรียนไม่อยู่ในช่üง 45- 61 กิโลกรัม ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 3) มีนักเรียนกี่เปอร์เซ็นต์ที่มีน้ำĀนักน้อยกü่า 40 กิโลกรัม ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 4) ถ้าดüงดาüมีน้ำĀนักเท่ากับเปอร์เซ็นไทล์ที่ 79.1 แล้üดüงดาüมีน้ำĀนักกี่กิโลกรัม ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. PIX > 481 Plz > 98 - so , a PIZJ - D. 51 plz < D.5) = D. 6915 1- P(45 < ✗ < 611 7- Pc 95 - so cz (61-50) e e 1 - Pl - 1.25 C Z C 2.751 I - 10.9970 - 0. 2056 ) I - D. 8974 = D. 1086 P ( ✗ cool plz c 402501 Pl 2<-2.51 =D. 0062×100 = 0.62-1. PC 2<01 ) = D. 7970 a = D- 81 ✗ - SO = D. 81 a ✗ = 3.24 1- SO ✗ = 53.24 kg
Āน่üยการเรียนรู้ที่ 2 การแจกแจงคüามน่าจะเป็น 59 8. คะแนนÿอบüิชาคณตýาÿตร์ของนักเรียนชั้นมัธยมýึกþาปีที่ 6 จำนüน 500 คน มีการแจกแจงปกติ ค่าเฉลี่ยเลขคณิตและÿ่üนเบี่ยงเบนมาตรฐานเป็น 58 คะแนน และ 7 คะแนน ตามลำดับแพรüาและพีระเป็น นักเรียนในกลุ่มนี้ แพรüาÿอบได้ 72 คะแนน และพีระÿอบได้ 75.5 คะแนนอยากทราบü่า มีนักเรียนประมาณกี่ คนที่ÿอบได้คะแนนมากกü่าแพรüาแต่น้อยกü่าพีระ ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 9. ขนาดของกระโปรงผู้ĀญิงกำĀนดโดยคüามยาüรอบเอü(นิ้ü) ถ้าคüามยาüรอบเอüมีการแจกแจงปกติ โดยมี คüามยาüรอบเอüเฉลี่ย 34 นิ้ü และÿ่üนเบี่ยงเบนมาตรฐาน 5 นิ้ü ซึ่งผู้ขายมีขนาดของกระโปรงผู้Āญิง เป็นดังนี้ ขนาดกระโปรง S M L XL คüามยาüรอบเอü(นิ้ü) 25-26 27-28 29-30 31-32 ใĀ้Āา 1) คüามน่าจะเป็นที่ลูกค้าไม่ÿามารถซื้อกระโปรงได้ตามขนาดที่ต้องการ ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 2) คüามน่าจะเป็นที่ลูกค้าต้องการซื้อกระโปรงขนาด M ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. 3) ถ้ามีลูกค้ามา 10 คน ใĀ้Āาคüามน่าจะเป็นที่มีเพียง 4 คนเท่านั้นที่ต้องการซื้อกระโปรงขนาด M ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. P ( 72 CX < 75.51 P 172-58<2 (75.5-58) 7 z P 12<2 C 2.51 = 0.9938 - D. 9772 = D. 0166 % = D. 0166 ✗ 100 = 2.66-1. War . = 1.66 × 500 700 = 8.30 = 8 AW 1- Pl 25 < ✗ < 32 ) = I - D. 3087 1- P( 25-34<2 (32-34) = 0.6913 s s 7- PC -7.8<2 C- D. 4) 7- ( D. 3446 - D. 03591 P( 274×428 ) =D . 775 - D. 0808 P( 27-39 C Z C 28-341 = 0,0343 Pl7.{ < 2<-1.2) S 10 ! 10.03937417-0.039316 416 ! = 270 ( D. 03431410.96571 = D. 000236