Answer:
d) The highest probability occurs when x equals 0.7500
Step-by-step explanation:
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability, given by the following formula:
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
In which
is the number of different combinations of x objects from a set of n elements, given by the following formula.
![C_{n,x} = \frac{n!}{x!(n-x)!}](https://tex.z-dn.net/?f=C_%7Bn%2Cx%7D%20%3D%20%5Cfrac%7Bn%21%7D%7Bx%21%28n-x%29%21%7D)
And p is the probability of X happening.
The expected value of the binomial distribution is:
![E(X) = np](https://tex.z-dn.net/?f=E%28X%29%20%3D%20np)
The standard deviation of the binomial distribution is:
![\sqrt{V(X)} = \sqrt{np(1-p)}](https://tex.z-dn.net/?f=%5Csqrt%7BV%28X%29%7D%20%3D%20%5Csqrt%7Bnp%281-p%29%7D)
In this problem, we have that:
![n = 15, p = \frac{1}{20}](https://tex.z-dn.net/?f=n%20%3D%2015%2C%20p%20%3D%20%5Cfrac%7B1%7D%7B20%7D)
a) The standard deviation is 0.8441
![\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{15*0.05*0.95} = 0.8441](https://tex.z-dn.net/?f=%5Csqrt%7BV%28X%29%7D%20%3D%20%5Csqrt%7Bnp%281-p%29%7D%20%3D%20%5Csqrt%7B15%2A0.05%2A0.95%7D%20%3D%200.8441)
This is correct
b) The number of trials is equal to 15
n is the number of trials and
. So this option is correct.
c) The probability that x equals 1 is 0.3658
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 1) = C_{15,1}.(0.05)^{1}.(0.95)^{14} = 0.3658](https://tex.z-dn.net/?f=P%28X%20%3D%201%29%20%3D%20C_%7B15%2C1%7D.%280.05%29%5E%7B1%7D.%280.95%29%5E%7B14%7D%20%3D%200.3658)
This option is correct.
d) The highest probability occurs when x equals 0.7500
False. The number of sucesses is a discrete number, that is, 0, 1, 2,...,15. P(X = 0.75), for example, does not exist.
e) The mean equals 0.7500
![E(X) = np = 15*0.05 = 0.75](https://tex.z-dn.net/?f=E%28X%29%20%3D%20np%20%3D%2015%2A0.05%20%3D%200.75)
This option is correct.
f) None of the above
d is false