Answer the following:
(A) Find the binomial probability P(x = 4), where n = 12 and p = 0.30.
(B) Set up, without solving, the binomial probability P(x is at most 4) using probability notation.
(C) How would you find the normal approximation to the binomial probability P(x = 4) in part A? Please show how you would calculate µ and σ in the formula for the normal approximation to the binomial, and show the final formula you would use without going through all the calculations.
=0.2311
(B) p(x is at most 4)=P (x,=4)=p(0)+p(1)+p(2)+p(3)+p(4)
=c(12,0)(0.30)^0(1-0.30)^2(12-0)+c(12,1)(0.30)^1(1-0.30)^(12-1)
+c(12,2)(0.30)^2(1-0.30)^(12-2)+c(12,3)(0.30)^3(1-0.30)^(12-3)+c(12,4)(0.30)^4(1-0.30)^(12-4)
(C) u=np=12*0.3=3.6
12*0.3*0.7=2.52
(2.52)=1.5875
to find p(4), we applies the continuity correction factor and find p(3.5<x<4.5). this because using the normal distribution p(x=4) will be 0.
z=(3.4-3.6)/1.5875=-0.0630and z=(4.5-3.6)/1.5875=0.5669
P(4)= P(3.5<x<4.5)=P(-0.063<z<0.5669)=0.2397
(A) To find the binomial probability P(x = 4), where n = 12 and p = 0.30, we can use the formula for binomial probability:
P(x) = (nCx) * p^x * (1-p)^(n-x)
where nCx represents "n choose x", and it can be calculated using the formula:
nCx = n! / (x! * (n-x)!)
In this case, n = 12, x = 4, and p = 0.30. Let's calculate P(x = 4):
P(4) = (12C4) * (0.30^4) * (0.70^8)
Calculating the values:
12C4 = 12! / (4! * (12-4)!) = 495
P(4) = 495 * (0.30^4) * (0.70^8) = 0.152
So, P(x = 4) is approximately 0.152.
(B) To set up the binomial probability P(x is at most 4) using probability notation, we can write:
P(x ≤ 4) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4)
This represents the sum of the probabilities of all possible outcomes from x = 0 to x = 4.
(C) To find the normal approximation to the binomial probability P(x = 4) in part A, we can use the formula for the mean (µ) and standard deviation (σ) of a binomial distribution, which are given by:
µ = n * p
σ = √(n * p * (1-p))
In this case, n = 12 and p = 0.30. Let's calculate µ and σ:
µ = 12 * 0.30 = 3.6
σ = √(12 * 0.30 * (1-0.30)) ≈ 1.799
Now, we can use these values to find the normal approximation for P(x = 4) using the formula for the normal distribution:
P(x = 4) ≈ P(3.5 < x < 4.5)
So, the final formula for the normal approximation to the binomial probability P(x = 4) is:
P(3.5 < x < 4.5) = Φ((4.5 - µ) / σ) - Φ((3.5 - µ) / σ)
where Φ represents the cumulative distribution function of the standard normal distribution.