Question

The probability is

Use the normal approximation to the binomial to find the probability for n-50, p 0.6, and X = 31, Round~-value calculations to 2 decimal places and final answer to 4 decimal places. The probability is

Answer 1

$\dpi{100} \fn_cm \mathbf{Solution}$

$\dpi{100} \fn_cm \mathbf{given:}p=0.6,q=0.4,n=50$

$\dpi{100} \fn_cm np=50*0.6=30\geq 5$

$\dpi{100} \fn_cm nq=50*0.4=20\geq 5$

$\dpi{100} \fn_cm \therefore\; Binomial \;random \;variable\; is \;approximately \;normal$

$\dpi{100} \fn_cm Mean\;(\mu)=np=30$

$\dpi{100} \fn_cm Standard\;deviation\;(\mu)=\sqrt{npq}=\sqrt{50*0.6*0.4}=\sqrt{12}=3.4641$

$\dpi{100} \fn_cm P(X=31)=P(30.5<X<31.5)\;(continuity \;correction)$

$\dpi{100} \fn_cm =P\left ( \frac{30.5-30}{3.4641}<\frac{X-\mu}{\sigma}<\frac{31.5-30}{3.4641} \right )\Rightarrow P(0.14<z<0.43)$

$\dpi{100} \fn_cm =P(0.43)-P(0.14)$

$\dpi{100} \fn_cm =0.6664-0.5557$

$\dpi{100} \fn_cm ={\color{Red} 0.1107}$