Normal Distribution: Finding Probabilities Silent Teacher Intelligent Practice Narration Your Turn 𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 85<𝑋<115 =𝑝 𝑃 𝑋<110 =𝑝 Practice

𝑃 85<𝑋<115 =𝑝 Lower: Upper: 𝜎: 𝜇: 𝑃 𝑋<110 =𝑝 Worked Examples Your Turn 𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 85<𝑋<115 =𝑝 Lower: Upper: 𝜎: 𝜇: 𝑃 𝑋<110 =𝑝 𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 92<𝑋<107 =𝑝 Lower: Upper: 𝜎: 𝜇: 𝑃 𝑋<123 =𝑝

𝑃 95<𝑋<105 =𝑝 𝑃 95<𝑋<105 =𝑝 𝑃 90<𝑋<110 =𝑝 𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝 𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐹𝑜𝑟 𝑋~𝑁 100,20 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 95<𝑋<105 =𝑝 𝑃 95<𝑋<105 =𝑝 𝑃 90<𝑋<110 =𝑝 𝑃 90<𝑋<110 =𝑝 𝑃 90<𝑋<100 =𝑝 𝑃 50<𝑋<100 =𝑝 𝑃 50<𝑋<100 =𝑝 𝐹𝑜𝑟 𝑋~𝑁 90,20 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 40<𝑋<100 =𝑝 𝑃 95<𝑋<105 =𝑝 𝑃 80<𝑋<140 =𝑝 𝑃 90<𝑋<110 =𝑝 𝑃 100<𝑋<160 =𝑝

𝑃 95<𝑋<105 =𝑝 𝑃 95<𝑋<105 =𝑝 𝑃 90<𝑋<110 =𝑝 𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝 𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐹𝑜𝑟 𝑋~𝑁 100,20 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 95<𝑋<105 =𝑝 𝑃 95<𝑋<105 =𝑝 𝑝=0.3829 𝑝=0.1974 𝑃 90<𝑋<110 =𝑝 𝑃 90<𝑋<110 =𝑝 𝑝=0.6827 𝑝=0.3829 𝑃 90<𝑋<100 =𝑝 𝑃 50<𝑋<100 =𝑝 𝑝=0.3413 𝑝=0.4937 𝑃 50<𝑋<100 =𝑝 𝐹𝑜𝑟 𝑋~𝑁 90,20 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑝=0.5000 𝑃 40<𝑋<100 =𝑝 𝑃 95<𝑋<105 =𝑝 𝑝=0.5000 𝑝=0.1746 𝑃 80<𝑋<140 =𝑝 𝑃 90<𝑋<110 =𝑝 𝑝=9772 𝑝=0.3413 𝑃 100<𝑋<160 =𝑝 𝑝=0.5000

𝑃 𝑋<105 =𝑝 𝑃 𝑋<105 =𝑝 𝑃 𝑋<110 =𝑝 𝑃 𝑋<90 =𝑝 𝑃 𝑋<150 =𝑝 𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝 𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐹𝑜𝑟 𝑋~𝑁 100,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 𝑋<105 =𝑝 𝑃 𝑋<105 =𝑝 𝑃 𝑋<110 =𝑝 𝑃 𝑋<90 =𝑝 𝑃 𝑋<150 =𝑝 𝑃 𝑋<75 =𝑝 𝐹𝑜𝑟 𝑋~𝑁 100,100 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐹𝑜𝑟 𝑋~𝑁 105,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 𝑋<105 =𝑝 𝑃 𝑋<105 =𝑝 𝑃 𝑋<90 =𝑝 𝑃 𝑋<110 =𝑝 𝑃 𝑋≤90 =𝑝 𝑃 𝑋<130 =𝑝

𝑃 𝑋<105 =𝑝 𝑃 𝑋<105 =𝑝 𝑃 𝑋<110 =𝑝 𝑃 𝑋<90 =𝑝 𝑃 𝑋<150 =𝑝 𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝 𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐹𝑜𝑟 𝑋~𝑁 100,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 𝑋<105 =𝑝 𝑃 𝑋<105 =𝑝 𝑝=0.6915 𝑝=0.8413 𝑃 𝑋<110 =𝑝 𝑃 𝑋<90 =𝑝 𝑝=0.8413 𝑝=0.0227 𝑃 𝑋<150 =𝑝 𝑃 𝑋<75 =𝑝 𝑝=1.0000 𝑝=0.0000 𝐹𝑜𝑟 𝑋~𝑁 100,100 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐹𝑜𝑟 𝑋~𝑁 105,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 𝑋<105 =𝑝 𝑃 𝑋<105 =𝑝 𝑝=0.6915 𝑝=0.5000 𝑃 𝑋<90 =𝑝 𝑃 𝑋<110 =𝑝 𝑝=0.1587 𝑝=0.8413 𝑃 𝑋≤90 =𝑝 𝑃 𝑋<130 =𝑝 𝑝=0.1587 𝑝=1.0000

𝑃 𝑋>105 =𝑝 𝑃 𝑋>105 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋≥95 =𝑝 𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝 𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐹𝑜𝑟 𝑋~𝑁 100,35 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 𝑋>105 =𝑝 𝑃 𝑋>105 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋≥95 =𝑝 𝑃 𝑋>64 =𝑝 𝐹𝑜𝑟 𝑋~𝑁 100,25 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐹𝑜𝑟 𝑋~𝑁 95,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 𝑋>105 =𝑝 𝑃 𝑋>105 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋≥80 =𝑝 𝑃 𝑋>100 =𝑝

𝑃 𝑋>105 =𝑝 𝑃 𝑋>105 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋≥95 =𝑝 𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝 𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐹𝑜𝑟 𝑋~𝑁 100,35 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 𝑋>105 =𝑝 𝑃 𝑋>105 =𝑝 𝑝=0.3085 𝑝=0.1990 𝑃 𝑋<95 =𝑝 𝑃 𝑋<95 =𝑝 𝑝=0.3085 𝑝=0.1990 𝑃 𝑋≥95 =𝑝 𝑃 𝑋>64 =𝑝 𝑝=0.6915 𝑝=1.0000 𝐹𝑜𝑟 𝑋~𝑁 100,25 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐹𝑜𝑟 𝑋~𝑁 95,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝑃 𝑋>105 =𝑝 𝑃 𝑋>105 =𝑝 𝑝=0.1587 𝑝=0.0227 𝑃 𝑋<95 =𝑝 𝑃 𝑋<95 =𝑝 𝑝=0.1587 𝑝=0.5000 𝑃 𝑋≥81 =𝑝 𝑃 𝑋>100 =𝑝 𝑝=0.9999 𝑝=0.1587