With n=13 and p= 0.7, find the binomial
probability p(9) by using a binomial
probability table. If np> and nq> 5, also
estimate the indicated probability by using the normal distribution as an
approximation to the binomial, if np<5 or
nq<5 then state that the normal approximation cannot be used.
A few hints:
If you use the normal distribution as an approximation, then you will need to find the mean and standard deviation.
mean = np = 13 * 0.7 = 9.1
standard deviation = √npq = √(13)(0.7)(0.3) = 1.65 (rounded)
Note: q = 1 - p
Next step: use z-scores
z = (x - mean)/sd
If x = 9, then
z = (9 - 9.1)/(1.65) = -0.06
Use a z-table to find the probability using the z-score.
To find the binomial probability, p(9), using a binomial probability table, we need to know the values for n and p. In this case, n = 13 and p = 0.7.
1. Using the Binomial Probability Table:
Step 1: Identify the values for n and p.
n = 13
p = 0.7
Step 2: Look for the row in the table that corresponds to n = 13.
Locate the row labeled "n = 13" in the binomial probability table.
Step 3: Find the column in the table that corresponds to p = 0.7.
Look for the column labeled "p = 0.7" in the same row you found in Step 2.
Step 4: Identify the probability value in the table at the intersection of the row and column found in Steps 2 and 3.
The value obtained from the table for p(9) with n = 13 and p = 0.7 will be your answer.
2. Using the Normal Approximation to the Binomial (if np>5 and nq>5):
To determine if we can use the normal approximation, we need to check if np and nq are both greater than 5. Here, np = 13 * 0.7 = 9.1, and nq = 13 * (1 - 0.7) = 3.9.
Since both np and nq are greater than 5, we can use the normal approximation.
Steps to estimate the probability using the normal approximation:
Step 1: Calculate the mean (μ) and standard deviation (σ) using the formulas:
μ = np = 13 * 0.7 = 9.1
σ = √(npq) = √(13 * 0.7 * 0.3) = √2.73 ≈ 1.65
Step 2: Convert the binomial probability problem to a normal probability problem. Since p(9) is a discrete probability, we need to apply continuity correction. We will calculate the probability of a range instead.
p(8.5 ≤ x ≤ 9.5)
Step 3: Standardize the range by converting it to a z-score using the formula:
z = (x - μ) / σ
Step 4: Use the z-table or a calculator to find the probability associated with the standardized range obtained in Step 3. The difference between the cumulative probabilities gives us the estimated probability.
If np<5 or nq<5, we cannot use the normal approximation.
So, in this case, we can use the normal approximation to estimate the probability p(9) by calculating the range probability p(8.5 ≤ x ≤ 9.5) using the z-table or a calculator.