1) Let "X=" the number of students who will drop out during or at the end of the 1st year
"X\\sim B(n,p)" Given that "p=0.25,n=1200." Then
"np=1200(0.25)=300>10,""n(1-p)=1200(1-0.25)=900>10" We can use a Normal Distribution to approximate a Binomial Distribution
"X^*\\sim N(\\mu=np,\\sigma^2=np(1-p))""\\mu=np=1200(0.25)=300,""\\sigma^2=np(1-p)=1200(0.25)(1-0.25)=225" "X^*\\sim N(300,15^2)"
Then
"Z={X^*-\\mu \\over \\sigma}\\sim N(0, 1)"Find the probability that more than 500 students will drop out during or at the end of the 1st year
"P(X>500)=P\\big(Z>{500-300 \\over 15}\\big)=1-P(Z\\leq{40 \\over 3})\\approx""\\approx7.4\\times10^{-41}\\approx0"
2) Let "X=" the number of patients experiencing side effects.
"X\\sim B(n,p)" Given that "p=0.07,n=300." Then
"np=300(0.07)=21\\geq10,""n(1-p)=300(1-0.07)=279\\geq10" We can use a Normal Distribution to approximate a Binomial Distribution
"X^*\\sim N(\\mu=np,\\sigma^2=np(1-p))""\\mu=np=300(0.07)=21,""\\sigma^2=np(1-p)=300(0.07)(1-0.07)=19.53" "X^*\\sim N(21,19.53)"
Then
"Z={X^*-\\mu \\over \\sigma}\\sim N(0, 1)" Find the probability that 50 and below of the 300 patients experience side effect
"P(X\\leq50)=P\\big(Z\\leq{50-21 \\over \\sqrt{19.53}}\\big)\\approx""\\approx P(Z\\leq6.562161)\\approx0.999999999973\\approx1"
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