Respuesta :
Answer:
[tex]P(X=20)=(60C20)(0.4)^{20} (1-0.4)^{60-20}=0.0616[/
[tex]\mu =np= 60*0.4= 24[/tex]
[tex]\sigma =\sqrt{60*0.4*(1-0.4)}= 3.79[/tex]
And using the normal approximation we have:
[tex] P(19.5 < X< 20.5)[/tex]
And we can use the z score formula and we got:
[tex] z=\frac{19.5-24}{3.79}= -1.19[/tex]
[tex] z=\frac{20.5-24}{3.79}= -0.92[/tex]
And using the normal standard distribution table we got:
[tex] P(-1.19<z<-0.92) =P(z<-0.92) -P(z<-1.19) =0.179-0.117= 0.062[/tex]
So we see that the results are very similar
Step-by-step explanation:
Let X the random variable of interest, on this case we now that:
[tex]X \sim Binom(n=60, p=0.4)[/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]
And we want to find this probability:
[tex] P(X=20)[/tex]
And we can find the probability using the probability mass function
[tex]P(X=20)=(60C20)(0.4)^{20} (1-0.4)^{60-20}=0.0616[/tex]
Using the normal approximation we need to find the mean and the deviation:
[tex]\mu =np= 60*0.4= 24[/tex]
[tex]\sigma =\sqrt{60*0.4*(1-0.4)}= 3.79[/tex]
And using the normal approximation we have:
[tex] P(19.5 < X< 20.5)[/tex]
And we can use the z score formula and we got:
[tex] z=\frac{19.5-24}{3.79}= -1.19[/tex]
[tex] z=\frac{20.5-24}{3.79}= -0.92[/tex]
And using the normal standard distribution table we got:
[tex] P(-1.19<z<-0.92) =P(z<-0.92) -P(z<-1.19) =0.179-0.117= 0.062[/tex]
So we see that the results are very similar
