บทที่ 2 การแจกแจงปกติ

เอกสารประกอบการสอน วิชาคณิตศาสตรเสริม 6 รหัสวชิ า ค33202 บทที่ 2 การแจกแจงปกติ Normal distribution จดั ทาํ โดย ครสู มุ าลี มสี กุล ตาํ แหนงครู วิทยฐานะชํานาญการพิเศษ กลมุ สาระการเรียนรูคณิตศาสตร โรงเรียนวเิ ศษไชยชาญ “ตันติวิทยาภมู ”ิ

หน่วยที 2 การแจกแจงปกติ 2.1 คะแนนมาตรฐาน (Standard Score) คะแนนมาตรฐานเป็นการปรับค่าของขอ้ มูลหรือคะแนนทีอยตู่ ่างกลุ่มกนั ใหม้ าอย่บู นมาตรวดั เดียวกนั ทาํ ใหข้ อ้ มูลทีไดห้ ลงั จากการปรับใหเ้ ป็ นคะแนนมาตรฐานสามารถนาํ มาเปรียบเทียบกนั ได้ สูตร คะแนนมาตรฐานของ xi คือ ……………………………………………………………………………….. ……………………………………………………………………………….. ตัวอย่าง จงหาคะแนนมาตรฐานของขอ้ มูลตอ่ ไปนี 1) คะแนนสอบ (xi) 45 คะแนน ค่าเฉลียเลขคณิต 30 คะแนน ส่วนเบียงเบนมาตรฐาน 5 คะแนน วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 2) นาํ หนกั (xi) 40.5 กิโลกรัม ค่าเฉลียเลขคณิต 48 กิโลกรัม ส่วนเบียงเบนมาตรฐาน 3 กิโลกรัม วธิ ีทํา................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 1

ตัวอย่าง จงหาค่าของ X จากสูตรของคะแนนมาตรฐาน เมือ 1) Z = 1.2 ค่าเฉลียเลขคณิต 30 ส่วนเบียงเบนมาตรฐาน วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 2) Z = –2 ค่าเฉลียเลขคณิต 15 ส่วนเบียงเบนมาตรฐาน 1.5 วธิ ีทํา................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... ตัวอย่าง สมพรสอบไดค้ ะแนนวชิ าภาษาองั กฤษในชนั ม. และ ม. เป็น คะแนน และ คะแนน ตามลาํ ดบั ถา้ ค่าเฉลียเลขคณิตและส่วนเบียงเบนมาตรฐานของคะแนนวชิ าภาษาองั กฤษของนกั เรียนทุกคน ในชนั ม. ทีสมพรเรียนอยเู่ ป็น คะแนน และ คะแนน และค่าเฉลียเลขคณิตและส่วนเบียงเบนมาตรฐาน ของคะแนนวชิ าภาษาองั กฤษของนกั เรียนทุกคนในชนั ม. เป็น คะแนน และ คะแนน ตามลาํ ดบั สมพร เรียนวชิ าภาษาองั กฤษในชนั ไหนไดด้ ีกวา่ กนั วธิ ีทํา................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 2

.......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... ตัวอย่าง ในการทดสอบเวลาทีใชใ้ นการแข่งรถของนกั แข่งระดบั ประเทศ เพือคดั เลือกตวั แทนไปแข่งกบั ประเทศอืน กาํ หนดไวว้ า่ ผทู้ ีผา่ นการทดสอบตอ้ งไดค้ ะแนนมาตรฐานของเวลาทีใชไ้ มม่ ากกวา่ . ถา้ ผลการ ทดสอบปรากฏว่านกั แข่งรถทีใชเ้ วลาในการขบั รถมากกวา่ นาที ไมผ่ า่ นการทดสอบ อยากทราบวา่ ในการ ทดสอบครังนีค่าเฉลยี เลขคณิตของเวลาทีใชใ้ นการแข่งรถของนกั แข่งทงั หมดเป็นเทา่ ไร ถา้ ส่วนเบียงเบน มาตรฐานของเวลาทีใชท้ งั หมดเป็น . นาที วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... ตัวอย่าง ในการสอบวดั ความสามารถในการใชภ้ าษาองั กฤษเพือการศึกษาต่อในระดบั อุดมศึกษา (CU-TEP) สุจติ ราสอบได้ คะแนน และดาํ รงสอบได้ คะแนน ถา้ คะแนนมาตรฐานของสุจิตราและดาํ รงเป็น และ . ตามลาํ ดบั จงหาค่าเฉลียเลขคณิตและส่วนเบียงเบนมาตรฐานของคะแนนสอบครังนี วธิ ีทํา................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 3

.......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 2.2 การแจกแจงปกตแิ ละเส้นโค้งปกติ (Normal distribution & Normal curve) 2 เสน้ โคง้ ปกติเป็นเสน้ โคง้ ทีกาํ หนดดว้ ยสมการ y = f(x) = 1  e  1  x 2 2   ทงั สองขา้ งของเสน้ โคง้ มีลกั ษณะสมมาตรกนั 4

ตารางแสดงพืนทีใต้เส้ นโค้ งปกติ 0z …………………………………………………………. …………………………………………………………. …………………………………………………………. …………………………………………………………. .......................................................................................................................................................................... .......................................................................................................................................................................... 5

ตัวอย่าง จงหาพืนทีใตโ้ คง้ ปกติจากค่า Z ทีกาํ หนดใหต้ ่อไปนี 1) 0  Z  1.27 วธิ ีทํา................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 2) –2.35  Z  0 วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 3) Z  2.14 วธิ ีทํา................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 4) Z  –0.99 วธิ ีทํา................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 6

5) –0.94  Z  1.37 วธิ ีทํา................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 6) 1.03  Z  2.11 วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... ตัวอย่าง ถา้ ขอ้ มลู ชุดหนึงมีการแจกแจงแบบปกติโดยมีคา่ เฉลียเลขคณิตเป็น 200 หน่วย และส่วนเบียงเบน มาตรฐานเป็ น 50 หน่วย อยากทราบว่ามีกีเปอร์เซน็ ตข์ องขอ้ มลู ซึงมีค่า 1) มากกวา่ 314 วธิ ีทํา................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 7

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.......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 2) นอ้ ยกว่า 126 วธิ ีทาํ 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.......................................................................................................................................................................... .......................................................................................................................................................................... 3) มากกวา่ 73 วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... 8

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.......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 4) นอ้ ยกว่า 297 วธิ ีทาํ ................................................................................................................................................................. 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ตัวอย่าง คะแนนสอบความถนดั ทวั ไป (General Attitude Test) สาํ หรับกลุม่ นกั เรียนหญงิ มคี ่าเฉลียเลขคณิต เป็น 120 คะแนน ส่วนเบียงเบนมาตรฐานเป็น 30 คะแนน และกลุ่มนกั เรียนชายมีค่าเฉลียเลขคณิตเป็น 100 คะแนน ส่วนเบียงเบนมาตรฐานเป็น 40 คะแนน ถา้ คะแนนของกลุ่มมีการแจกแจงปกติ อยากทราบวา่ 1) ถา้ น.ส.ป๋ ุย สอบได้ 130 คะแนน คะแนนของเธอเป็นตาํ แหน่งเปอร์เซ็นไทลท์ ีเทา่ ไรของคะแนนในกลุ่ม นกั เรียนหญิง วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 2) ถา้ นายกอลฟ์ สอบได้ 110 คะแนน คะแนนของเขาเป็ นตาํ แหน่งเปอร์เซน็ ไทลท์ ีเท่าไรของคะแนนใน กลุ่มนกั เรียนชาย และเป็นตาํ แหน่งเปอร์เซ็นไทลท์ ีเทา่ ไรของคะแนนในกลมุ่ นกั เรียนหญิง วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 10

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.......................................................................................................................................................................... ตัวอย่าง ในการผลิตลวดทองแดงของผผู้ ลิตรายหนึงปรากฏวา่ ขนาดเสน้ ผา่ นศูนยก์ ลางของลวดมีการแจก แจงแบบปกติโดยมีขนาดเฉลีย 0.1125 เซนติเมตร ความแปรปรวนเป็น 0.00000144 เซนติเมตร2 จงหาวา่ ลวดทองแดงทีผลิตไดซ้ ึงมีความหนาอยรู่ ะหว่าง 0.1120 และ 0.1135 เซนติเมตร มีกีเปอร์เซน็ ต์ วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... 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.......................................................................................................................................................................... .......................................................................................................................................................................... ตัวอย่าง ปริมาตรสุทธิของแชมพูทีผลติ โดยบริษทั แห่งหนึงมีการแจกแจงแบบปกติ โดยมีปริมาตรสุทธิเฉลีย เป็น 250 ซีซี ถา้ ขวดทีมีปริมาตรสุทธินอ้ ยกวา่ 235 ซีซี มีอยู่ 10.56% จงหาความแปรปรวนของปริมาตรสุทธิ ขวดบรรจุแชมพทู ีผลิตโดยบริษทั นี วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... 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.......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... ตัวอย่าง ให้ x เป็นความคลาดเคลือนของเวลาทีเกิดขึนจากนาฬิกาทีผลิตโดยโรงงานแห่งหนึง ถา้ ความคลาด เคลือนมีการแจกแจงปกติและมีค่าเฉลียเลขคณิต 0.00 วนิ าที ความแปรปรวน 0.25 วนิ าที2 อยากทราบวา่ x ซึงทาํ ให้ 66.45% ของนาฬิกาทงั หมดทีผลิตได้ มีความคลาดเคลือนระหวา่ ง x กบั 0.55 วินาที วธิ ีทาํ ................................................................................................................................................................. .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 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.......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... ตัวอย่าง การแจกแจงของคะแนนสอบครังหนึงเป็ นการแจกแจงปกติ โดยมีค่าเฉลียเลขคณิต 75 คะแนน และ ส่วนเบียงเบนมาตรฐาน 15 คะแนน จงหา 1) คะแนนทีเป็นเปอร์เซ็นตไ์ ทลท์ ี 45 วธิ ีทาํ ................................................................................................................................................................. 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.......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... .......................................................................................................................................................................... 2) คะแนนทีเป็นเปอร์เซ็นตไ์ ทลท์ ี 80 วธิ ีทาํ 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