Answer:
a)
b)
And adding these values we got:
![P(36 \leq X \leq 38)= 0.1535](https://tex.z-dn.net/?f=%20P%2836%20%5Cleq%20X%20%5Cleq%2038%29%3D%200.1535)
c) We can find the expected value given by:
![E(X) = np =50*0.65 = 32.5](https://tex.z-dn.net/?f=%20E%28X%29%20%3D%20np%20%3D50%2A0.65%20%3D%2032.5)
And the standard deviation would be:
![\sigma = \sqrt{np(1-p)} \sqrt{50*0.65*(1-0.65)}= 3.373](https://tex.z-dn.net/?f=%20%5Csigma%20%3D%20%5Csqrt%7Bnp%281-p%29%7D%20%5Csqrt%7B50%2A0.65%2A%281-0.65%29%7D%3D%203.373)
We can use the approximation to the normal distribution and we have at leat 95% of the data within 2 deviations from the mean. And the lower limit for this case would be:
![\mu -2\sigma = 32.5- 2*3.373 = 25.75](https://tex.z-dn.net/?f=%20%5Cmu%20-2%5Csigma%20%3D%2032.5-%202%2A3.373%20%3D%2025.75)
And then we can consider a value of 18 as unusual lower for this case.
Step-by-step explanation:
Let X the random variable of interest "number cleared by arrest or exceptional", on this case we can model this variable with this distribution:
The probability mass function for the Binomial distribution is given as:
Where (nCx) means combinatory and it's given by this formula:
Part a
We want this probability:
Part b
We want this probability:
![P(36 \leq X \leq 38)](https://tex.z-dn.net/?f=%20P%2836%20%5Cleq%20X%20%5Cleq%2038%29)
We can find the individual probabilities:
And adding these values we got:
![P(36 \leq X \leq 38)= 0.1535](https://tex.z-dn.net/?f=%20P%2836%20%5Cleq%20X%20%5Cleq%2038%29%3D%200.1535)
Part c
We can find the expected value given by:
![E(X) = np =50*0.65 = 32.5](https://tex.z-dn.net/?f=%20E%28X%29%20%3D%20np%20%3D50%2A0.65%20%3D%2032.5)
And the standard deviation would be:
![\sigma = \sqrt{np(1-p)} \sqrt{50*0.65*(1-0.65)}= 3.373](https://tex.z-dn.net/?f=%20%5Csigma%20%3D%20%5Csqrt%7Bnp%281-p%29%7D%20%5Csqrt%7B50%2A0.65%2A%281-0.65%29%7D%3D%203.373)
We can use the approximation to the normal distribution and we have at leat 95% of the data within 2 deviations from the mean. And the lower limit for this case would be:
![\mu -2\sigma = 32.5- 2*3.373 = 25.75](https://tex.z-dn.net/?f=%20%5Cmu%20-2%5Csigma%20%3D%2032.5-%202%2A3.373%20%3D%2025.75)
And then we can consider a value of 18 as unusual lower for this case.