91Ó°ÊÓ

1% of the patients who undergo laser surgery to correct the vision have serious postlaser vision problem.

Let x be the number of patients undergo the laser surgery to correct the vision who have the serious post laser problem.

With n =100,000 and p =0.001

Hence,

$$\begin{gathered} E\left(x\right)=\mathrm{μ} \\ =np \\ =100,000×0.01 \\ E\left(x\right)=1000 \end{gathered}$$

Hence mean is 1000.

$\begin{array}{l}\mathrm{Î\pm }=\sqrt{npq}\\ =\sqrt{100,000×0.01-0.01)}\\ =\sqrt{1000,000×0.01×0.99}\\ \mathrm{σ}=31.646\end{array}$

Hence the value of standard deviation is 31.646

Most of the observations will fall within three standard deviation of the mean.

That is, atleast $\frac{8}{9}$of the measurement will fall within 3 standard deviation of the mean.

Lets $\mathrm{μ}=1000$and $\mathrm{μ}=31.464$

The observations will fall within three standard deviation of the mean is,

$\mathrm{μ}Â\pm 3\mathrm{σ}=1000Â\pm 3\left(31.464\right)$

$$\begin{gathered} =\left(1000-3\left(31.464\right),1000+3\left(31.464\right)\right) \\ =\left(1000-94.392,1000+3\left(31.464\right)\right) \end{gathered}$$

$\mathrm{μ}Â\pm 3\mathrm{σ}=\left(905,608,1,094.392\right)$

Hence, the normal approximation is appropriate because the interval$\left(905,608,1,094.392\right)$ is lies in the range of 0 to 1000,000

The approximate probability can be obtained by using the formula

$z=\frac{\left(k-0.5\right)-\mathrm{μ}}{\mathrm{σ}}$

Therefore,

$$\begin{gathered} \mathrm{P}\left(\mathrm{x}<950\right)=\mathrm{P}\left(\mathrm{z}<\frac{\left(950-0.5\right)-\mathrm{μ}}{\mathrm{σ}}\right) \\ =\mathrm{P}\left(\mathrm{z}<\frac{\left(949.5-1000\right)}{31.464}\right) \\ \mathrm{P}\left(\mathrm{x}<950\right)=\mathrm{P}\left(\mathrm{z}<\frac{\left(949.5-1000\right)}{31.464}\right) \\ =\mathrm{P}\left(\mathrm{z}<\frac{-50.5}{31.464}\right) \end{gathered}$$

$P\left(z≤-1.61\right)$[from table 2-normal curve areas]

$\begin{array}{l}=0.5-0.4463\\ =0.0537\end{array}$

Hence, approximate probability that fewer than 950 patients will experience serious post laser vision problem is 0.0537