Respuesta :
Answer:
a) [tex]P(X=0)=(4C0)(0.75)^0 (1-0.75)^{4-0}=0.0039[/tex]
[tex]P(X=1)=(4C1)(0.75)^1 (1-0.75)^{4-1}=0.0469[/tex]
[tex]P(X=2)=(4C2)(0.75)^2 (1-0.75)^{4-2}=0.211[/tex]
[tex]P(X=3)=(4C3)(0.75)^2 (1-0.75)^{4-3}=0.422[/tex]
[tex]P(X=4)=(4C4)(0.75)^2 (1-0.75)^{4-4}=0.316[/tex]
b) [tex] E(X) = np = 4*0.75=3[/tex]
c) [tex] Sd(X) =\sqrt{np(1-p)}=\sqrt{4*0.75*(1-0.75)}=0.866[/tex]
d) [tex] P(X \geq 3) \geq 0.98[/tex]
And the dsitribution that satisfy this is [tex]X\sim Binom(n=9,p=0.75[/tex]
Step-by-step explanation:
Previous concepts
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
Let X the random variable of interest, on this case we now that:
[tex]X \sim Binom(n=4, p=1-0.25=0.75)[/tex]
The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
Part a
[tex]P(X=0)=(4C0)(0.75)^0 (1-0.75)^{4-0}=0.0039[/tex]
[tex]P(X=1)=(4C1)(0.75)^1 (1-0.75)^{4-1}=0.0469[/tex]
[tex]P(X=2)=(4C2)(0.75)^2 (1-0.75)^{4-2}=0.211[/tex]
[tex]P(X=3)=(4C3)(0.75)^2 (1-0.75)^{4-3}=0.422[/tex]
[tex]P(X=4)=(4C4)(0.75)^2 (1-0.75)^{4-4}=0.316[/tex]
Part b
The expected value is givn by:
[tex] E(X) = np = 4*0.75=3[/tex]
Part c
For the standard deviation we have this:
[tex]Sd(X) =\sqrt{np(1-p)}=\sqrt{4*0.75*(1-0.75)}=0.866[/tex]
Part d
For this case the sample size needs to be higher or equal to 9. Since we need a value such that:
[tex] P(X \geq 3) \geq 0.98[/tex]
And the dsitribution that satisfy this is [tex]X\sim Binom(n=9,p=0.75[/tex]
We can verify this using the following code:
"=1-BINOM.DIST(3,9,0.75,TRUE)" and we got 0.99 and the condition is satisfied.
