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弌-亳亶仆 舒仍舒 舒仆亞亳亶仆 舒亳亳从亳亶仆 亰舒亳仄 舒亞仍亞舒

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弌舒亳亳从亳亶仆 亰舒亳仄 仂亶仍亞仂仍亞 亰舒舒, 弍仂亟仍仂亞仂 弍仂亟仂 舒亞舒 亰勵亶 /5 亟舒 亢亳仍亟 舒亢亳仍仍舒亢 弍亶 仄舒亠仄舒亳从亳亶仆 弍舒亞亳亶仆 勵仆亟仆 亞舒仍/ -亳亶仆 丐-仆 仄亞亢亳仍仆, 舒亞舒 亰勵亶 .舒于舒舒勵仆
仍亶 仍仍勵勵仍亞: 仍 1. 舒亞舒 弍仂仍仂于仂仍仆 仄舒亠仄舒亳从亳亶仆 唏唏仍弍唏 弍亶 舒亳亳从 (唏亞唏亞亟唏仍亶 舒亢亳仍仍舒)-亳亶仆 亞仂仍 舒亞亟舒仆, 仂亶仍亞仂仍亟亞 仆仍仆 勵勵? 亅亟亞 仆 弍仂仍仂仆 舒仍舒 舒仆亞亳亶仆 舒亳亳从亳亶仆 舒亞仍亞舒舒亶 仆 仂仍弍仂亞亟仂 于? /) _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ ____________________________________ 仍 2. 仍舒 舒仆亞亳亶仆 亞舒仍仆 唏唏仍弍唏 亞舒亞亟舒仆 舒亳亳从亳亶仆 舒亞亟舒仆, 仆 仂仄仂仆亟舒舒 亰舒舒舒亟 勵仆亟仍亶 弍ﾑ, 唏仄仆唏 唏唏仍弍唏唏唏 舒亳舒仆 勵于仆亳亶 舒亞仍亞亟亞 仆仍仆 勵勵. _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _____________________
亞仍亞舒 1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 亳仂亞舒仄 丶亞仆 亟亳舒亞舒仄 弌舒舒仍舒亶 舒亶舒亞 仆舒于仆 亟亳舒亞舒仄 2. 舒亳仄 舒亳亳从 勵亰勵勵仍仍勵勵亟 /唏于亳亶仆 舒仆亟仍舒亞仆 弍舒 弍舒亶仍仆 仄亢亳亞亟勵勵仆, 亟亠亳仍, 从于舒亳仍, 舒仆亟舒 舒亰舒亶仍, 亟亳仗亠 / 3. 勵仍亞仍仆 唏亞唏亞亟仍亳亶仆 亟仆亟舒亢, 仄亠亟亳舒仆, 仄仂仂亟 仂仍仂 舒亞舒 4. 丱于亳亶仆 舒舒仍, 于亳亶仆 舒舒仍仆 勵仆亞 舒亳亞仍舒仆 仄舒亞舒亟仍舒仍 仂仍仂 舒亞舒
1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 丱亳仄仍舒亞亟舒仆 亟舒仄舒仄亢亳亶仆 勵仆亞 弍舒 亞舒亳从: 弌舒 仄亢 亳亶仆 丱亳仄仍舒仍 3- 舒 $120 $120 4- 舒 $50 $170 5- 舒 $110 $280 6- 舒 $100 $380 7- 舒 $50 $430 8- 舒 $20 $450
1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 -仆舒于仆 亟亳舒亞舒仄 /Stem and Leaf Plots/ ∃∃夷術亳仄 亟舒 弍勵 勵仄勵勵亳亶仆 唏仆亟唏 弍舒 亞仍仆 仄亢亞 亳 仆舒于仆 亟亳舒亞舒仄舒舒 亟勵仍亠 ∃ 仆舒于仆 亟亳舒亞舒仄 舒亳亞仍舒 仆 礆舒 舒 仂仍弍仂亞亟仂仍仂亶 于? ∃ 仆舒于仆 亟亳舒亞舒仄 舒亳亞仍舒仆 亟仆亟舒亢, 仄亠亟亳舒仆, 仄仂仂亟, 亟舒仍舒亶亞 仂仍亞仂仂.
1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 仆舒于仆 亟亳舒亞舒仄舒舒 亟勵仍仆 唏亞唏亞亟仍唏唏 舒仄亳 舒亟舒亞 弍舒 舒亟舒亞亞勵亶 勵仄勵勵亳亶仆 仆 亟舒舒仍仆 亟舒仍舒亶, 仄仂仂亟, 仄亠亟亳舒仆, 亟仆亟舒亢 亰亞亳亶亞 仂仍亢 舒仍舒舒舒亶. 亅仆 亟舒仍亞舒舒仆亟 亟仆 勵仆 舒仄舒亞亟舒仆 弍舒亶仆舒 于? -仆舒于仆 亟亳舒亞舒仄 /Stem and Leaf Plots/ 丱丕 丱丕 舒仍舒亶 仂仂亟 亠亟亳舒仆 亳仄亠亳从 亟仆亟舒亢 丐舒仄亳 舒亟舒亞 丐舒仄亳 舒亟舒亞亞勵亶
1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 丶亞仆 亟亳舒亞舒仄 丱舒仆亟仍舒亞仆 仍仆 /亠亞亠亳亶仆 仍仆/ 仂亠仍亳亶仆 从仂亳亳亠仆 仆 2 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆亳亶 仂仂仂仆亟仆 舒仄舒舒仍仆 亰亞亳亶亞 仂亟仂仂亶仍仆仂.
1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 仂亠仍 /舒仄舒舒仍亞 仆仍亶/ A B C D
仂亠仍亳亶仆 舒仆舒亟 唏仆亟唏-y (仄) 亢亳仆-x (从亞) y-y x-x (y-y)^2 (x-x)^2 (y-y)*(x-x) a1 150 55.0 150.0 55.0 22,500 3,025 8,250.0 a2 120 38.0 120.0 38.0 14,400 1,444 4,560 a3 89 18.0 89.0 18.0 7,921 324 1,602 a4 165 54.0 165.0 54.0 27,225 2,916 8,910 a5 173 67.0 173.0 67.0 29,929 4,489 11,591 a6 180 95.0 180.0 95.0 32,400 9,025 17,100 a7 151 45.0 151.0 45.0 22,801 2,025 6,795 a8 165 61.0 165.0 61.0 27,225 3,721 10,065 a9 120 40.0 120.0 40.0 14,400 1,600 4,800 a10 190 85.0 190.0 85.0 36,100 7,225 16,150 1,503 558 1,503 558 234,901 234,901 89,823 r (仂亠仍亳亶仆 从仂亳亳亠仆) 0.9796 r^2 (亠亠仄亳仆舒亳亶仆 亰亞) 0.9596 - 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 - 50 100 150 200 AxisTitle Axis Title
弌舒舒仍舒亶 舒亶舒亞仆 亟亳舒亞舒仄 弌舒舒仍舒亶 舒亶舒亞仆 亟亳舒亞舒仄仄 /Box and Whisker Plot/ 弌舒舒仍舒亶 舒亶舒亞仆 亟亳舒亞舒仄仄 /"Box and Whisker Plot"/-舒舒 仄舒 仂仍仂仆 仄亟仍仍亳亶亞 仂仍亢 舒舒 弍仂仍仂仄亢仂亶. 亳 1. 舒舒舒 唏亞唏亞亟仍亳亶仆 亢舒亞舒舒仍亞 舒舒仍舒亶 舒亶舒亞仆 亟亳舒亞舒仄仄舒舒 亟勵仍亢, 从于舒亳仍 仂仂仂仆亟仆 亟舒仍舒亶亞 仂仍. 4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11 亠亠仆亳仍: 哦亞唏亞亟仍亳亶仆 弍舒亶仍亞 亳仍亳亶仍 仄亢亳亞亟勵勵仆. /亰舒亳仄亟舒舒 仗仂亠仆舒仍 亞亢 弍亳仆 弍舒亶亟舒亞./ 丐舒亶仆 唏亞唏亞亟仍唏唏 弍舒亞舒 亞舒舒亶 唏亞唏亞亟唏仍 仆亳亶 唏亞唏亞亟仍亳亶仆 亟仆 于亳亶亞 亰仍亢 弍舒亶亞舒舒亞 亳仍亳亶仍仆. 亳 仆: 20 勵仆亳亶 亟仂仂 舒 唏仆亟唏唏唏唏 4- 亢舒亞舒仆 亞亠. 丐亞于仍 80% 仆 舒仆舒舒 仆舒仄舒仆 勵仄勵勵 弍舒亶仆舒 弍ﾑ 舒仆 唏仆亟亳亶仆 勵亰勵勵仍仍 80% 亠亠仆亳仍 /仗仂亠仆舒仍/-舒亶 弍舒亶仆舒 亞仆 勵亞. 哦唏唏唏 仍弍仍 "1.85仄" 亞仆 唏亞唏亞亟仍亳亶仆 于亟 仗亠亠仆亳仍 仆 80 弍舒亶仆舒 亞亢 亟舒亞. 亠亳仍 /Deciles/: 哦亞唏亞亟仍勵勵亟亳亶仆 仄弍仍亞亟仆 亢舒亞舒舒仍亞 10; 10%-亳舒 弍勵仍亞勵勵亟亟 于舒舒亢 勵亰仆亳亶亞 亠亳仍 /Deciles/ 亞亟亞. 亳 仆 2- 亟亠亳仍 仆 20%-亳亶仆 仗亠亠仆亳仍亶 亟舒于舒仆舒
舒于舒仄亢亳亶仆 勵仆亞 唏亞唏亞亟仍亳亶仆 亟仆亟舒亢 亞舒, 仄仂仂亟, 仄亠亟亳舒仆 仂仍仂 x f f*x 3 2 6 8 7 56 13 8 104 18 3 54 亳亶仍弍: 20 220 舒于舒仄亢亳亶仆 勵仆亞 舒亳亞仍舒仆 仄仂仂亟, 仄亠亟亳舒仆, 亟仆舒亢, 亟舒仍舒亶 亰亞亳亶亞 仂仍仂仂仂亶
勵仍亞仍仆 唏亞唏亞亟仍亳亶仆 亟仆亟舒亢 x f f*x 3 4 12 8 6 48 13 8 104 18 2 36 亳亶仍弍: 20 200 丐仂亞仍仂仂仄仆 仂仂 舒于舒仄亢 1 - 5 4 6 - 10 6 11 - 15 8 16 - 20 2 勵仍亞 弍勵亳亶仆 仂仂仆亟 勵仆亳亶 亟仆亟舒亢 亞亳亶仆 弍舒亶仍仍舒仆 亟舒舒舒 勵仆亞亳亶亞 亳亶亢 弍仂仍仂 ﾐ /
勵仍亞仍仆 唏亞唏亞亟仍亳亶仆 仄仂仂亟, 仄亠亟亳舒仆亞 仂仍仂 勵仆亞 亞亳仂亞舒仄 弍舒亶亞仍亢, 亟仆亟舒亢, 仄仂仂亟, 仄亠亟亳舒仆亞 仂仍仂仂仂亶. 9 6 73 2
舒亞舒仍 1. 哦仄仆唏 亞仍仍舒仆 唏亞唏亞亟仍唏唏 舒亳亞仍舒仆 舒仆亳仍仆 亞仆 亟舒亳亞舒仄 弍舒亶亞仍舒舒舒亶 2. 仂仂亟, 仄亠亟亳舒仆, 亟仆亟舒亢 唏仆亟唏亳亶亞 仂仍仂仂仂亶. 3. 哦仆亟唏 弍舒 亢亳仆亞亳亶仆 舒仄舒舒仍亞 亳仍亳亶仍 从仂亠仍亳亶亞 弍仂亟仂亢 勵亰亶. 4. 仆舒于仆 亟亳舒亞舒仄舒舒 唏仆亟唏亳亶仆 于亟 从于舒亳仍亟亞 仂仍. 5. 亟仂仂 勵勵仆亳亶亞 舒舒仍舒亶 舒亶舒亞仆 亟亳舒亞舒仄舒舒 亳仍亳亶仍亠. 5. 哦仆亟唏亳亶仆 仄亢亞 5 唏亞唏仆亶 亞亳仂亞舒仄舒舒 亟勵仍亶. 6. 亳仂亞舒仄 舒亳亞仍舒仆 弍勵仍亞仍仆 唏亞唏亞亟仍亳亶仆 勵仆仍仆 仄仂仂亟, 仄亠亟亳舒仆, 亟仆亟舒亢 亰亞亳亶亞 弍仂亟仂亢, 舒仆亟舒亞 唏亞唏亞亟仍唏唏 仂仍仂仆 勵亰勵勵仍仍勵勵亟亶 舒仍亢 亳仍.
弌舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆 亳从亠 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆 丐舒舒仍亞勵亶 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆 亳从亠 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆亳亶 舒舒仍仆 仆从 弌舒仆亟舒 舒亰舒亶仍 弍舒 亟亳仗亠 亳仆仂仄 舒舒仍 丱于亳亶仆 舒舒仍
3. 弌舒仆亟舒 舒亰舒亶仍 弌舒仆亟舒 舒亰舒亶仍 仆 勵勵于亳亶仆 亞亟 亟仆亟亢舒舒舒舒 仂仍 亰舒亶亟 舒舒亢 弍舒亶亞舒舒亞 舒仍亟舒亞. 丐勵勵仆亳亶亞 勵亞 仄亟亞仍仆.. 弌舒仆亟舒 舒亰舒亶仍 仆 仆亞亳亶仆 仂仄仂亞仂仂 弍仂亟仂亞亟仂 弍唏亞唏唏亟 亟亳仗亠亳亶仆 舒亳仄亠亳从 从于舒亟舒 磶亞舒亶 仆勵勵. 丐亞于仍 亟亳仗亠 亞亢 ﾑ 于? 亳仗亠 (variance) 亞亟亞 仆 亟仆亟亢舒舒 舒亰舒亶 舒亰舒亶仍亟仆 从于舒亟舒仍舒亞 亟仆亟舒亢 ﾐ. 亳仗亠亳亶亞 仂仍仂亟仂仂 亟舒舒舒 亟舒舒舒仍仍舒舒 仂仍仆仂. 1.丐勵勵于亳亶仆 弍勵 亞亟仆 亟仆亟亢亳亶亞 仂仍仆仂. 2.丐勵勵于亳亶仆 弍勵 亞亟仆 亟仆亟亢舒舒 舒亰舒亶 舒亰舒亶仍亟仆 从于舒亟舒亟亞 仂仍仆仂. 哦唏唏 仍弍仍 亞舒 弍勵 亟亟仆亢亳亶亞 舒 磿亞舒于亞 从于舒亟舒 亰亞 亟于勵勵仍仆. 3.丐亟亞 磿亞舒于仆 从于舒亟舒亟仆 亟仆亟亢亳亶亞 仂仍仆仂.
Basic of statistics
弌舒仆亟舒 舒亰舒亶仍 弍舒 亟仆亟亢舒舒 仄亶 仆 舒仄舒舒舒 于?
弌舒仆亟舒 舒亰舒亶仍 弍舒 仄亶仆 仍弍
丱于亳亶仆 舒舒仍仆 仍 m - 仄舒亠仄舒亳从 亟仆亟舒亢 (亰舒亳仄亟舒舒 袖 于仍 a-舒舒 仄亟亞仍 仆 弍亳亶) - 舒仆亟舒 舒亰舒亶仍m
丱于亳亶仆 舒舒仍仆 仄亶仆 仆亳亶 舒仍弍舒亶 1 弍舒亶 仍: 亅仆亟 仂亞仄仂仍亞 仂仍弍仂仍: 哦唏唏唏 仍弍仍 仆仂仄舒仍 舒舒仍舒亶 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆亳亶 舒舒仍仆 仆从 仆 /舒亶仆 仄舒亞舒亟仍舒仍 仆/ 弌舒仆亟舒 舒亰舒亶仍 弍舒 仄舒亠仄舒亳从 亟仆亟舒亢 亞仆 仂 仗舒舒仄亠 舒仄舒舒仆舒. 丱于 舒舒仍亞勵亶 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆亳亶 舒舒仍仆 仆从 仆 亟舒舒舒 仍弍亶 弍仂仍 于亳亶仆 舒舒仍舒亶 弍舒亶仆舒 亞亟亞. 勳勵仆亟: 丱于亳亶仆 舒舒仍仆 亟 2 仗舒舒仄亠亳亶亞 舒亳亞仍舒仆 仄亟亞仍仆.
弌舒仆亟舒 于亳亶仆 舒舒仍: 弍舒亶 舒舒仍亞 舒仆亟舒 于亳亶仆 舒舒仍 亞亟亞. 弌舒仆亟舒 于亳亶仆 舒舒仍仆 勵仆亞亳亶亞 舒亳亞仍舒仆 Z 仂仆仂仂仆 仍舒仄亢舒亶亞舒舒 弍舒亟 于亳亶仆 舒舒仍舒亶 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆亳亶 仄舒亞舒亟仍舒仍亞 仂仍亟仂亞. z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
Let us use table of normal distibution and Statistics Calculator
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
亢亳仆亳亶 于亟 舒仆舒仄舒亞勵亶 仂仆亞仂仂仆 舒亞 80- 舒舒 弍舒亞舒亞勵亶 仂仆仂仂 舒于舒仆 弍舒亶 仄舒亞舒亟仍舒仍 亟 弍舒亶 于? 仂亟仂亢 仂仍仂仂仂亶. ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ 弌舒仆舒仄舒亞勵亶 仂仆亞仂仂仆 舒亞 80-90 仂仆仂仂 舒于舒仆 弍舒亶 仄舒亞舒亟仍舒仍亞 仂仂仂仂仍亢 仂仍仂仂仂亶. ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ________________________________________________
1.X is a normally normally distributed variable with mean 亮 = 30 and standard deviation = 4. Find a) P(x < 40) b) P(x > 21) c) P(30 < x < 35) 2.A radar unit is used to measure speeds of cars on a motorway. The speeds are normally distributed with a mean of 90 km/hr and a standard deviation of 10 km/hr. What is the probability that a car picked at random is travelling at more than 100 km/hr? 3.For a certain type of computers, the length of time bewteen charges of the battery is normally distributed with a mean of 50 hours and a standard deviation of 15 hours. John owns one of these computers and wants to know the probability that the length of time will be between 50 and 70 hours. 4.Entry to a certain University is determined by a national test. The scores on this test are normally distributed with a mean of 500 and a standard deviation of 100. Tom wants to be admitted to this university and he knows that he must score better than at least 70% of the students who took the test. Tom takes the test and scores 585. Will he be admitted to this university?
5. The length of similar components produced by a company are approximated by a normal distribution model with a mean of 5 cm and a standard deviation of 0.02 cm. If a component is chosen at random a) what is the probability that the length of this component is between 4.98 and 5.02 cm? b) what is the probability that the length of this component is between 4.96 and 5.04 cm?
Answers to the Above QuestionsNote: What is meant here by area is the area under the standard normal curve. a) For x = 40, the z-value z = (40 - 30) / 4 = 2.5 Hence P(x < 40) = P(z < 2.5) = [area to the left of 2.5] = 0.9938 b) For x = 21, z = (21 - 30) / 4 = -2.25 Hence P(x > 21) = P(z > -2.25) = [total area] - [area to the left of -2.25] = 1 - 0.0122 = 0.9878 c) For x = 30 , z = (30 - 30) / 4 = 0 and for x = 35, z = (35 - 30) / 4 = 1.25 Hence P(30 < x < 35) = P(0 < z < 1.25) = [area to the left of z = 1.25] - [area to the left of 0] = 0.8944 - 0.5 = 0.3944
2. Let x be the random variable that represents the speed of cars. x has 亮 = 90 and = 10. We have to find the probability that x is higher than 100 or P(x > 100) For x = 100 , z = (100 - 90) / 10 = 1 P(x > 90) = P(z >, 1) = [total area] - [area to the left of z = 1] = 1 - 0.8413 = 0.1587 The probability that a car selected at a random has a speed greater than 100 km/hr is equal to 0.1587
3. Let x be the random variable that represents the length of time. It has a mean of 50 and a standard deviation of 15. We have to find the probability that x is between 50 and 70 or P( 50< x < 70) For x = 50 , z = (50 - 50) / 15 = 0 For x = 70 , z = (70 - 50) / 15 = 1.33 (rounded to 2 decimal places) P( 50< x < 70) = P( 0< z < 1.33) = [area to the left of z = 1.33] - [area to the left of z = 0] = 0.9082 - 0.5 = 0.4082 The probability that John's computer has a length of time between 50 and 70 hours is equal to 0.4082.
4. Let x be the random variable that represents the scores. x is normally ditsributed with a mean of 500 and a standard deviation of 100. The total area under the normal curve represents the total number of students who took the test. If we multiply the values of the areas under the curve by 100, we obtain percentages. For x = 585 , z = (585 - 500) / 100 = 0.85 The proportion P of students who scored below 585 is given by P = [area to the left of z = 0.85] = 0.8023 = 80.23% Tom scored better than 80.23% of the students who took the test and he will be admitted to this University.
5. a) P(4.98 < x < 5.02) = P(-1 < z < 1) = 0.6826 b) P(4.96 < x < 5.04) = P(-2 < z < 2) = 0.9544
7. What length of time marks the shortest 70% of all pregnancies? Normal Distribution 袖 = 266 = 16 P(X < ?) = 0.70 P(Z < ?) = 0.70 Z = 0.52 X = 266 + 0.52(16) X = 266 + 8.32 X = 274.32
For each question, construct a normal distribution curve and label the horizontal axis. Then answer each question. The mean life of a tire is 30 000 km. The standard deviation is 2000 km. 1. 68% of all tires will have a life between ______km and ______km. 2. 95% of all tires will have a life between ______km and ______km. 3. What percent of the tires will have a life that exceeds 26000 km? 4. If a company purchased 2000 tires, how many tires would you expect to last more than 28 000 km?
亳仆仂仄 舒舒仍 弍舒 于亳亶仆 舒舒仍
亳仆仂仄 舒舒仍 弍舒 于亳亶仆 舒舒仍
Basic of statistics
Basic of statistics

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Basic of statistics

  • 1. 弌舒亳亳从亳亶仆 亰舒亳仄 仂亶仍亞仂仍亞 亰舒舒, 弍仂亟仍仂亞仂 弍仂亟仂 舒亞舒 亰勵亶 /5 亟舒 亢亳仍亟 舒亢亳仍仍舒亢 弍亶 仄舒亠仄舒亳从亳亶仆 弍舒亞亳亶仆 勵仆亟仆 亞舒仍/ -亳亶仆 丐-仆 仄亞亢亳仍仆, 舒亞舒 亰勵亶 .舒于舒舒勵仆
  • 2. 仍亶 仍仍勵勵仍亞: 仍 1. 舒亞舒 弍仂仍仂于仂仍仆 仄舒亠仄舒亳从亳亶仆 唏唏仍弍唏 弍亶 舒亳亳从 (唏亞唏亞亟唏仍亶 舒亢亳仍仍舒)-亳亶仆 亞仂仍 舒亞亟舒仆, 仂亶仍亞仂仍亟亞 仆仍仆 勵勵? 亅亟亞 仆 弍仂仍仂仆 舒仍舒 舒仆亞亳亶仆 舒亳亳从亳亶仆 舒亞仍亞舒舒亶 仆 仂仍弍仂亞亟仂 于? /) _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ ____________________________________ 仍 2. 仍舒 舒仆亞亳亶仆 亞舒仍仆 唏唏仍弍唏 亞舒亞亟舒仆 舒亳亳从亳亶仆 舒亞亟舒仆, 仆 仂仄仂仆亟舒舒 亰舒舒舒亟 勵仆亟仍亶 弍ﾑ, 唏仄仆唏 唏唏仍弍唏唏唏 舒亳舒仆 勵于仆亳亶 舒亞仍亞亟亞 仆仍仆 勵勵. _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _____________________
  • 3. 亞仍亞舒 1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 亳仂亞舒仄 丶亞仆 亟亳舒亞舒仄 弌舒舒仍舒亶 舒亶舒亞 仆舒于仆 亟亳舒亞舒仄 2. 舒亳仄 舒亳亳从 勵亰勵勵仍仍勵勵亟 /唏于亳亶仆 舒仆亟仍舒亞仆 弍舒 弍舒亶仍仆 仄亢亳亞亟勵勵仆, 亟亠亳仍, 从于舒亳仍, 舒仆亟舒 舒亰舒亶仍, 亟亳仗亠 / 3. 勵仍亞仍仆 唏亞唏亞亟仍亳亶仆 亟仆亟舒亢, 仄亠亟亳舒仆, 仄仂仂亟 仂仍仂 舒亞舒 4. 丱于亳亶仆 舒舒仍, 于亳亶仆 舒舒仍仆 勵仆亞 舒亳亞仍舒仆 仄舒亞舒亟仍舒仍 仂仍仂 舒亞舒
  • 4. 1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 丱亳仄仍舒亞亟舒仆 亟舒仄舒仄亢亳亶仆 勵仆亞 弍舒 亞舒亳从: 弌舒 仄亢 亳亶仆 丱亳仄仍舒仍 3- 舒 $120 $120 4- 舒 $50 $170 5- 舒 $110 $280 6- 舒 $100 $380 7- 舒 $50 $430 8- 舒 $20 $450
  • 5. 1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 -仆舒于仆 亟亳舒亞舒仄 /Stem and Leaf Plots/ ∃∃夷術亳仄 亟舒 弍勵 勵仄勵勵亳亶仆 唏仆亟唏 弍舒 亞仍仆 仄亢亞 亳 仆舒于仆 亟亳舒亞舒仄舒舒 亟勵仍亠 ∃ 仆舒于仆 亟亳舒亞舒仄 舒亳亞仍舒 仆 礆舒 舒 仂仍弍仂亞亟仂仍仂亶 于? ∃ 仆舒于仆 亟亳舒亞舒仄 舒亳亞仍舒仆 亟仆亟舒亢, 仄亠亟亳舒仆, 仄仂仂亟, 亟舒仍舒亶亞 仂仍亞仂仂.
  • 6. 1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 仆舒于仆 亟亳舒亞舒仄舒舒 亟勵仍仆 唏亞唏亞亟仍唏唏 舒仄亳 舒亟舒亞 弍舒 舒亟舒亞亞勵亶 勵仄勵勵亳亶仆 仆 亟舒舒仍仆 亟舒仍舒亶, 仄仂仂亟, 仄亠亟亳舒仆, 亟仆亟舒亢 亰亞亳亶亞 仂仍亢 舒仍舒舒舒亶. 亅仆 亟舒仍亞舒舒仆亟 亟仆 勵仆 舒仄舒亞亟舒仆 弍舒亶仆舒 于? -仆舒于仆 亟亳舒亞舒仄 /Stem and Leaf Plots/ 丱丕 丱丕 舒仍舒亶 仂仂亟 亠亟亳舒仆 亳仄亠亳从 亟仆亟舒亢 丐舒仄亳 舒亟舒亞 丐舒仄亳 舒亟舒亞亞勵亶
  • 7. 1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 丶亞仆 亟亳舒亞舒仄 丱舒仆亟仍舒亞仆 仍仆 /亠亞亠亳亶仆 仍仆/ 仂亠仍亳亶仆 从仂亳亳亠仆 仆 2 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆亳亶 仂仂仂仆亟仆 舒仄舒舒仍仆 亰亞亳亶亞 仂亟仂仂亶仍仆仂.
  • 8. 1. 哦亞唏亞亟仍亳亶亞 亟勵仍 亰舒亳仄 舒亞舒 仂亠仍 /舒仄舒舒仍亞 仆仍亶/ A B C D
  • 9. 仂亠仍亳亶仆 舒仆舒亟 唏仆亟唏-y (仄) 亢亳仆-x (从亞) y-y x-x (y-y)^2 (x-x)^2 (y-y)*(x-x) a1 150 55.0 150.0 55.0 22,500 3,025 8,250.0 a2 120 38.0 120.0 38.0 14,400 1,444 4,560 a3 89 18.0 89.0 18.0 7,921 324 1,602 a4 165 54.0 165.0 54.0 27,225 2,916 8,910 a5 173 67.0 173.0 67.0 29,929 4,489 11,591 a6 180 95.0 180.0 95.0 32,400 9,025 17,100 a7 151 45.0 151.0 45.0 22,801 2,025 6,795 a8 165 61.0 165.0 61.0 27,225 3,721 10,065 a9 120 40.0 120.0 40.0 14,400 1,600 4,800 a10 190 85.0 190.0 85.0 36,100 7,225 16,150 1,503 558 1,503 558 234,901 234,901 89,823 r (仂亠仍亳亶仆 从仂亳亳亠仆) 0.9796 r^2 (亠亠仄亳仆舒亳亶仆 亰亞) 0.9596 - 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 - 50 100 150 200 AxisTitle Axis Title
  • 10. 弌舒舒仍舒亶 舒亶舒亞仆 亟亳舒亞舒仄 弌舒舒仍舒亶 舒亶舒亞仆 亟亳舒亞舒仄仄 /Box and Whisker Plot/ 弌舒舒仍舒亶 舒亶舒亞仆 亟亳舒亞舒仄仄 /"Box and Whisker Plot"/-舒舒 仄舒 仂仍仂仆 仄亟仍仍亳亶亞 仂仍亢 舒舒 弍仂仍仂仄亢仂亶. 亳 1. 舒舒舒 唏亞唏亞亟仍亳亶仆 亢舒亞舒舒仍亞 舒舒仍舒亶 舒亶舒亞仆 亟亳舒亞舒仄仄舒舒 亟勵仍亢, 从于舒亳仍 仂仂仂仆亟仆 亟舒仍舒亶亞 仂仍. 4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11 亠亠仆亳仍: 哦亞唏亞亟仍亳亶仆 弍舒亶仍亞 亳仍亳亶仍 仄亢亳亞亟勵勵仆. /亰舒亳仄亟舒舒 仗仂亠仆舒仍 亞亢 弍亳仆 弍舒亶亟舒亞./ 丐舒亶仆 唏亞唏亞亟仍唏唏 弍舒亞舒 亞舒舒亶 唏亞唏亞亟唏仍 仆亳亶 唏亞唏亞亟仍亳亶仆 亟仆 于亳亶亞 亰仍亢 弍舒亶亞舒舒亞 亳仍亳亶仍仆. 亳 仆: 20 勵仆亳亶 亟仂仂 舒 唏仆亟唏唏唏唏 4- 亢舒亞舒仆 亞亠. 丐亞于仍 80% 仆 舒仆舒舒 仆舒仄舒仆 勵仄勵勵 弍舒亶仆舒 弍ﾑ 舒仆 唏仆亟亳亶仆 勵亰勵勵仍仍 80% 亠亠仆亳仍 /仗仂亠仆舒仍/-舒亶 弍舒亶仆舒 亞仆 勵亞. 哦唏唏唏 仍弍仍 "1.85仄" 亞仆 唏亞唏亞亟仍亳亶仆 于亟 仗亠亠仆亳仍 仆 80 弍舒亶仆舒 亞亢 亟舒亞. 亠亳仍 /Deciles/: 哦亞唏亞亟仍勵勵亟亳亶仆 仄弍仍亞亟仆 亢舒亞舒舒仍亞 10; 10%-亳舒 弍勵仍亞勵勵亟亟 于舒舒亢 勵亰仆亳亶亞 亠亳仍 /Deciles/ 亞亟亞. 亳 仆 2- 亟亠亳仍 仆 20%-亳亶仆 仗亠亠仆亳仍亶 亟舒于舒仆舒
  • 11. 舒于舒仄亢亳亶仆 勵仆亞 唏亞唏亞亟仍亳亶仆 亟仆亟舒亢 亞舒, 仄仂仂亟, 仄亠亟亳舒仆 仂仍仂 x f f*x 3 2 6 8 7 56 13 8 104 18 3 54 亳亶仍弍: 20 220 舒于舒仄亢亳亶仆 勵仆亞 舒亳亞仍舒仆 仄仂仂亟, 仄亠亟亳舒仆, 亟仆舒亢, 亟舒仍舒亶 亰亞亳亶亞 仂仍仂仂仂亶
  • 12. 勵仍亞仍仆 唏亞唏亞亟仍亳亶仆 亟仆亟舒亢 x f f*x 3 4 12 8 6 48 13 8 104 18 2 36 亳亶仍弍: 20 200 丐仂亞仍仂仂仄仆 仂仂 舒于舒仄亢 1 - 5 4 6 - 10 6 11 - 15 8 16 - 20 2 勵仍亞 弍勵亳亶仆 仂仂仆亟 勵仆亳亶 亟仆亟舒亢 亞亳亶仆 弍舒亶仍仍舒仆 亟舒舒舒 勵仆亞亳亶亞 亳亶亢 弍仂仍仂 ﾐ /
  • 13. 勵仍亞仍仆 唏亞唏亞亟仍亳亶仆 仄仂仂亟, 仄亠亟亳舒仆亞 仂仍仂 勵仆亞 亞亳仂亞舒仄 弍舒亶亞仍亢, 亟仆亟舒亢, 仄仂仂亟, 仄亠亟亳舒仆亞 仂仍仂仂仂亶. 9 6 73 2
  • 14. 舒亞舒仍 1. 哦仄仆唏 亞仍仍舒仆 唏亞唏亞亟仍唏唏 舒亳亞仍舒仆 舒仆亳仍仆 亞仆 亟舒亳亞舒仄 弍舒亶亞仍舒舒舒亶 2. 仂仂亟, 仄亠亟亳舒仆, 亟仆亟舒亢 唏仆亟唏亳亶亞 仂仍仂仂仂亶. 3. 哦仆亟唏 弍舒 亢亳仆亞亳亶仆 舒仄舒舒仍亞 亳仍亳亶仍 从仂亠仍亳亶亞 弍仂亟仂亢 勵亰亶. 4. 仆舒于仆 亟亳舒亞舒仄舒舒 唏仆亟唏亳亶仆 于亟 从于舒亳仍亟亞 仂仍. 5. 亟仂仂 勵勵仆亳亶亞 舒舒仍舒亶 舒亶舒亞仆 亟亳舒亞舒仄舒舒 亳仍亳亶仍亠. 5. 哦仆亟唏亳亶仆 仄亢亞 5 唏亞唏仆亶 亞亳仂亞舒仄舒舒 亟勵仍亶. 6. 亳仂亞舒仄 舒亳亞仍舒仆 弍勵仍亞仍仆 唏亞唏亞亟仍亳亶仆 勵仆仍仆 仄仂仂亟, 仄亠亟亳舒仆, 亟仆亟舒亢 亰亞亳亶亞 弍仂亟仂亢, 舒仆亟舒亞 唏亞唏亞亟仍唏唏 仂仍仂仆 勵亰勵勵仍仍勵勵亟亶 舒仍亢 亳仍.
  • 16. 3. 弌舒仆亟舒 舒亰舒亶仍 弌舒仆亟舒 舒亰舒亶仍 仆 勵勵于亳亶仆 亞亟 亟仆亟亢舒舒舒舒 仂仍 亰舒亶亟 舒舒亢 弍舒亶亞舒舒亞 舒仍亟舒亞. 丐勵勵仆亳亶亞 勵亞 仄亟亞仍仆.. 弌舒仆亟舒 舒亰舒亶仍 仆 仆亞亳亶仆 仂仄仂亞仂仂 弍仂亟仂亞亟仂 弍唏亞唏唏亟 亟亳仗亠亳亶仆 舒亳仄亠亳从 从于舒亟舒 磶亞舒亶 仆勵勵. 丐亞于仍 亟亳仗亠 亞亢 ﾑ 于? 亳仗亠 (variance) 亞亟亞 仆 亟仆亟亢舒舒 舒亰舒亶 舒亰舒亶仍亟仆 从于舒亟舒仍舒亞 亟仆亟舒亢 ﾐ. 亳仗亠亳亶亞 仂仍仂亟仂仂 亟舒舒舒 亟舒舒舒仍仍舒舒 仂仍仆仂. 1.丐勵勵于亳亶仆 弍勵 亞亟仆 亟仆亟亢亳亶亞 仂仍仆仂. 2.丐勵勵于亳亶仆 弍勵 亞亟仆 亟仆亟亢舒舒 舒亰舒亶 舒亰舒亶仍亟仆 从于舒亟舒亟亞 仂仍仆仂. 哦唏唏 仍弍仍 亞舒 弍勵 亟亟仆亢亳亶亞 舒 磿亞舒于亞 从于舒亟舒 亰亞 亟于勵勵仍仆. 3.丐亟亞 磿亞舒于仆 从于舒亟舒亟仆 亟仆亟亢亳亶亞 仂仍仆仂.
  • 18. 弌舒仆亟舒 舒亰舒亶仍 弍舒 亟仆亟亢舒舒 仄亶 仆 舒仄舒舒舒 于?
  • 20. 丱于亳亶仆 舒舒仍仆 仍 m - 仄舒亠仄舒亳从 亟仆亟舒亢 (亰舒亳仄亟舒舒 袖 于仍 a-舒舒 仄亟亞仍 仆 弍亳亶) - 舒仆亟舒 舒亰舒亶仍m
  • 21. 丱于亳亶仆 舒舒仍仆 仄亶仆 仆亳亶 舒仍弍舒亶 1 弍舒亶 仍: 亅仆亟 仂亞仄仂仍亞 仂仍弍仂仍: 哦唏唏唏 仍弍仍 仆仂仄舒仍 舒舒仍舒亶 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆亳亶 舒舒仍仆 仆从 仆 /舒亶仆 仄舒亞舒亟仍舒仍 仆/ 弌舒仆亟舒 舒亰舒亶仍 弍舒 仄舒亠仄舒亳从 亟仆亟舒亢 亞仆 仂 仗舒舒仄亠 舒仄舒舒仆舒. 丱于 舒舒仍亞勵亶 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆亳亶 舒舒仍仆 仆从 仆 亟舒舒舒 仍弍亶 弍仂仍 于亳亶仆 舒舒仍舒亶 弍舒亶仆舒 亞亟亞. 勳勵仆亟: 丱于亳亶仆 舒舒仍仆 亟 2 仗舒舒仄亠亳亶亞 舒亳亞仍舒仆 仄亟亞仍仆.
  • 22. 弌舒仆亟舒 于亳亶仆 舒舒仍: 弍舒亶 舒舒仍亞 舒仆亟舒 于亳亶仆 舒舒仍 亞亟亞. 弌舒仆亟舒 于亳亶仆 舒舒仍仆 勵仆亞亳亶亞 舒亳亞仍舒仆 Z 仂仆仂仂仆 仍舒仄亢舒亶亞舒舒 弍舒亟 于亳亶仆 舒舒仍舒亶 舒仆舒仄舒亞勵亶 仄亢亳亞亟勵勵仆亳亶 仄舒亞舒亟仍舒仍亞 仂仍亟仂亞. z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
  • 23. Let us use table of normal distibution and Statistics Calculator
  • 24. z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
  • 25. 亢亳仆亳亶 于亟 舒仆舒仄舒亞勵亶 仂仆亞仂仂仆 舒亞 80- 舒舒 弍舒亞舒亞勵亶 仂仆仂仂 舒于舒仆 弍舒亶 仄舒亞舒亟仍舒仍 亟 弍舒亶 于? 仂亟仂亢 仂仍仂仂仂亶. ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ 弌舒仆舒仄舒亞勵亶 仂仆亞仂仂仆 舒亞 80-90 仂仆仂仂 舒于舒仆 弍舒亶 仄舒亞舒亟仍舒仍亞 仂仂仂仂仍亢 仂仍仂仂仂亶. ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ________________________________________________
  • 26. 1.X is a normally normally distributed variable with mean 亮 = 30 and standard deviation = 4. Find a) P(x < 40) b) P(x > 21) c) P(30 < x < 35) 2.A radar unit is used to measure speeds of cars on a motorway. The speeds are normally distributed with a mean of 90 km/hr and a standard deviation of 10 km/hr. What is the probability that a car picked at random is travelling at more than 100 km/hr? 3.For a certain type of computers, the length of time bewteen charges of the battery is normally distributed with a mean of 50 hours and a standard deviation of 15 hours. John owns one of these computers and wants to know the probability that the length of time will be between 50 and 70 hours. 4.Entry to a certain University is determined by a national test. The scores on this test are normally distributed with a mean of 500 and a standard deviation of 100. Tom wants to be admitted to this university and he knows that he must score better than at least 70% of the students who took the test. Tom takes the test and scores 585. Will he be admitted to this university?
  • 27. 5. The length of similar components produced by a company are approximated by a normal distribution model with a mean of 5 cm and a standard deviation of 0.02 cm. If a component is chosen at random a) what is the probability that the length of this component is between 4.98 and 5.02 cm? b) what is the probability that the length of this component is between 4.96 and 5.04 cm?
  • 28. Answers to the Above QuestionsNote: What is meant here by area is the area under the standard normal curve. a) For x = 40, the z-value z = (40 - 30) / 4 = 2.5 Hence P(x < 40) = P(z < 2.5) = [area to the left of 2.5] = 0.9938 b) For x = 21, z = (21 - 30) / 4 = -2.25 Hence P(x > 21) = P(z > -2.25) = [total area] - [area to the left of -2.25] = 1 - 0.0122 = 0.9878 c) For x = 30 , z = (30 - 30) / 4 = 0 and for x = 35, z = (35 - 30) / 4 = 1.25 Hence P(30 < x < 35) = P(0 < z < 1.25) = [area to the left of z = 1.25] - [area to the left of 0] = 0.8944 - 0.5 = 0.3944
  • 29. 2. Let x be the random variable that represents the speed of cars. x has 亮 = 90 and = 10. We have to find the probability that x is higher than 100 or P(x > 100) For x = 100 , z = (100 - 90) / 10 = 1 P(x > 90) = P(z >, 1) = [total area] - [area to the left of z = 1] = 1 - 0.8413 = 0.1587 The probability that a car selected at a random has a speed greater than 100 km/hr is equal to 0.1587
  • 30. 3. Let x be the random variable that represents the length of time. It has a mean of 50 and a standard deviation of 15. We have to find the probability that x is between 50 and 70 or P( 50< x < 70) For x = 50 , z = (50 - 50) / 15 = 0 For x = 70 , z = (70 - 50) / 15 = 1.33 (rounded to 2 decimal places) P( 50< x < 70) = P( 0< z < 1.33) = [area to the left of z = 1.33] - [area to the left of z = 0] = 0.9082 - 0.5 = 0.4082 The probability that John's computer has a length of time between 50 and 70 hours is equal to 0.4082.
  • 31. 4. Let x be the random variable that represents the scores. x is normally ditsributed with a mean of 500 and a standard deviation of 100. The total area under the normal curve represents the total number of students who took the test. If we multiply the values of the areas under the curve by 100, we obtain percentages. For x = 585 , z = (585 - 500) / 100 = 0.85 The proportion P of students who scored below 585 is given by P = [area to the left of z = 0.85] = 0.8023 = 80.23% Tom scored better than 80.23% of the students who took the test and he will be admitted to this University.
  • 32. 5. a) P(4.98 < x < 5.02) = P(-1 < z < 1) = 0.6826 b) P(4.96 < x < 5.04) = P(-2 < z < 2) = 0.9544
  • 33. 7. What length of time marks the shortest 70% of all pregnancies? Normal Distribution 袖 = 266 = 16 P(X < ?) = 0.70 P(Z < ?) = 0.70 Z = 0.52 X = 266 + 0.52(16) X = 266 + 8.32 X = 274.32
  • 34. For each question, construct a normal distribution curve and label the horizontal axis. Then answer each question. The mean life of a tire is 30 000 km. The standard deviation is 2000 km. 1. 68% of all tires will have a life between ______km and ______km. 2. 95% of all tires will have a life between ______km and ______km. 3. What percent of the tires will have a life that exceeds 26000 km? 4. If a company purchased 2000 tires, how many tires would you expect to last more than 28 000 km?
  • 35. 亳仆仂仄 舒舒仍 弍舒 于亳亶仆 舒舒仍
  • 36. 亳仆仂仄 舒舒仍 弍舒 于亳亶仆 舒舒仍