Answer the following:
(A) Find the binomial probability P(x = 5), where n = 12 and p = 0.70.
(B) Set up, without solving, the binomial probability P(x is at most 5) using probability notation.
(C) How would you find the normal approximation to the binomial probability P(x = 5) 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.
Question 11
Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. Round to three decimal places. n = 5, x = 2, p = 0.70
(A) To find the binomial probability P(x = 5) where n = 12 and p = 0.70, we can use the formula:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
where C(n, x) represents the binomial coefficient.
Plugging in the values, we have:
P(5) = C(12, 5) * 0.7^5 * (1-0.7)^(12-5)
(B) To set up the binomial probability P(x is at most 5) using probability notation, we need to sum up the probabilities from x = 0 to x = 5.
P(x ≤ 5) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5)
(C) To find the normal approximation to the binomial probability P(x = 5) in part (A), we first need to calculate µ (mean) and σ (standard deviation) using the formulas:
µ = n * p
σ = sqrt(n * p * (1-p))
Plugging in the values from part (A), we have:
µ = 12 * 0.70
σ = sqrt(12 * 0.70 * (1-0.70))
The final formula for the normal approximation to the binomial probability P(x = 5) is:
P(x = 5) ≈ P(4.5 < x < 5.5)
This can then be approximated using the standard normal distribution.
(A) To find the binomial probability P(x = 5), where n = 12 and p = 0.70, we can use the formula for the binomial probability:
P(x) = (nCx) * p^x * (1-p)^(n-x)
where nCx represents the number of ways to choose x items from a set of n items.
To calculate P(x = 5), we substitute the values into the formula:
P(x = 5) = (12C5) * (0.70)^5 * (1-0.70)^(12-5)
We can evaluate this expression to find the answer.
(B) The binomial probability P(x is at most 5) can be represented using probability notation as:
P(x ≤ 5)
This means we want to find the probability of getting a value of x that is less than or equal to 5.
(C) To find the normal approximation to the binomial probability P(x = 5) in part A, we can use the following steps:
1. Calculate the mean (µ) and standard deviation (σ) of the binomial distribution using the formulas:
µ = n * p
σ = √(n * p * (1-p))
2. Substitute the values of n and p from part A into the formulas to calculate µ and σ.
3. Use the normal approximation to the binomial distribution, which states that for large values of n, the binomial distribution can be approximated by a normal distribution with mean µ and standard deviation σ.
The final formula to calculate the normal approximation to the binomial probability P(x = 5) is:
P(x = 5) ≈ P(z = (5 - µ) / σ)
where z is the standardized value obtained by subtracting the mean µ from x (5 in this case) and dividing by the standard deviation σ.