Answer:
a. 0.7291
b. 0.9968
c. 0.7259
Step-by-step explanation:
a. np and n(1-p) can be calculated as:
![np=23\times 0.48\\\\=11.04\\\\n(1-p)=23(1-0.52)\\\\=11.96](https://tex.z-dn.net/?f=np%3D23%5Ctimes%200.48%5C%5C%5C%5C%3D11.04%5C%5C%5C%5Cn%281-p%29%3D23%281-0.52%29%5C%5C%5C%5C%3D11.96)
#Both np and np(1-p) are greater than 5, hence, normal approximation is most appropriate:
![\mu_x=11.04\\\\\sigma^2=np(1-p)=0.48\times 0.52\times 23=5.7408](https://tex.z-dn.net/?f=%5Cmu_x%3D11.04%5C%5C%5C%5C%5Csigma%5E2%3Dnp%281-p%29%3D0.48%5Ctimes%200.52%5Ctimes%2023%3D5.7408)
#Define Y:
Y~(11.04,5.7408)
![P(X\leq 12)\approx P(Y\leq 12.5)\\\\P(Z\leq \frac{(12.5-11.04)}{\sqrt{5.7408}})=\\\\=1-0.2709\\\\=0.7291](https://tex.z-dn.net/?f=P%28X%5Cleq%2012%29%5Capprox%20P%28Y%5Cleq%2012.5%29%5C%5C%5C%5CP%28Z%5Cleq%20%5Cfrac%7B%2812.5-11.04%29%7D%7B%5Csqrt%7B5.7408%7D%7D%29%3D%5C%5C%5C%5C%3D1-0.2709%5C%5C%5C%5C%3D0.7291)
Hence, the probability of 12 or fewer is 0.8291
b. The probability that 5 or more fish were caught.
#Using normal approximation:
![P(X \geq 5) \approx P(Y \geq 4.5) = P(Z \geq\frac{ (4.5-11.04)}{\sqrt{(5.7408)}})\\\\=1-0.0032\\\\=0.9968](https://tex.z-dn.net/?f=P%28X%20%5Cgeq%205%29%20%5Capprox%20P%28Y%20%5Cgeq%204.5%29%20%3D%20P%28Z%20%5Cgeq%5Cfrac%7B%20%284.5-11.04%29%7D%7B%5Csqrt%7B%285.7408%29%7D%7D%29%5C%5C%5C%5C%3D1-0.0032%5C%5C%5C%5C%3D0.9968)
Hence, the probability of catching 5+ is 0.9968
c. The probability of between 5 and 12 is calculated as;
-From b above
and a ,
=0.7291
![P(5\leq X\leq 120\approx P(4.5\leq Y\leq 12.5)\\\\=0.7291-(1-0.9968)\\\\=0.7259](https://tex.z-dn.net/?f=P%285%5Cleq%20X%5Cleq%20120%5Capprox%20P%284.5%5Cleq%20Y%5Cleq%20%2012.5%29%5C%5C%5C%5C%3D0.7291-%281-0.9968%29%5C%5C%5C%5C%3D0.7259)
Hence, the probability of between 5 and 12 is 0.7259