Answer:
Probability that more than eight but fewer than 12 of the 20 constituents sampled believe their representative possesses low ethical standards is 0.417890.
Step-by-step explanation:
We are given that the paper claims that 43% of all constituents believe their representative possesses low ethical standards.
Suppose 20 of a representative's constituents are randomly and independently sampled.
The above situation can be represented through binomial distribution;
![P(X=r) = \binom{n}{r} \times p^{r} \times (1-p)^{n-r} ; x = 0,1,2,3,.....](https://tex.z-dn.net/?f=P%28X%3Dr%29%20%3D%20%5Cbinom%7Bn%7D%7Br%7D%20%5Ctimes%20p%5E%7Br%7D%20%5Ctimes%20%281-p%29%5E%7Bn-r%7D%20%3B%20x%20%3D%200%2C1%2C2%2C3%2C.....)
where, n = number of trials (samples) taken = 20 constituents
r = number of success = more than eight but fewer than 12
p = probability of success which in our question is probability that
all constituents believe their representative possesses low
ethical standards, i.e; p = 43%
Let X = <u><em>Number of constituents who believe their representative possesses low ethical standards</em></u>
So, X ~ Binom(n = 20 , p = 0.43)
Now, Probability that more than eight but fewer than 12 of the 20 constituents sampled believe their representative possesses low ethical standards is given by = P(8 < X < 12)
P(8 < X < 12) = P(X = 9) + P(X = 10) + P(X = 11)
= ![\binom{20}{9} \times 0.43^{9} \times (1-0.43)^{20-9}+ \binom{20}{10} \times 0.43^{10} \times (1-0.43)^{20-10}+\binom{20}{11} \times 0.43^{11} \times (1-0.43)^{20-11}](https://tex.z-dn.net/?f=%5Cbinom%7B20%7D%7B9%7D%20%5Ctimes%200.43%5E%7B9%7D%20%5Ctimes%20%281-0.43%29%5E%7B20-9%7D%2B%20%5Cbinom%7B20%7D%7B10%7D%20%5Ctimes%200.43%5E%7B10%7D%20%5Ctimes%20%281-0.43%29%5E%7B20-10%7D%2B%5Cbinom%7B20%7D%7B11%7D%20%5Ctimes%200.43%5E%7B11%7D%20%5Ctimes%20%281-0.43%29%5E%7B20-11%7D)
=
= <u>0.417890</u>
<u></u>
Hence, the required probability is 0.417890.