Solution :
Given :
n = 200, p = 80% = 0.8, q = 1 - p = 0.2
[tex]$\mu = np$[/tex]
[tex]$= 200 \times 0.8 = 160$[/tex]
[tex]$\sigma= \sqrt{npq}$[/tex]
[tex]$\sigma= \sqrt{200 \times 0.8 \times 0.2}$[/tex]
= 5.6569
a). x = 169.5
∴ [tex]$z = \frac{x- \mu}{\sigma}$[/tex]
[tex]$z = \frac{169.5- 160}{5.6569}$[/tex]
= 1.6794
[tex]$P(x \geq 170) = P(z>1.6794) = 0.0465$[/tex]
Therefore, probability that the stolen goods were not recovered in 170 robberies or more is = 0.0465
b). x= 149.5
[tex]$z = \frac{x- \mu}{\sigma}$[/tex]
[tex]$z = \frac{149.5- 160}{5.6569} = -1.8561$[/tex]
[tex]$P(x \geq 150) = P(z>-1.8561) = 0.9683$[/tex]
Therefore the probability that the stolen goods were not recovered in the 150 or in more robberies is = 0.9683
