Consider a binomial experiment with 20 trials and probability 0.45 on a single trial. Use the normal distribution to find the probability of exactly 10 successes. Round your answer to the thousandths place.
Prob
= C(20,10)(.45^10)(.55^10)
= .15926
It's 0.162
To find the probability of exactly 10 successes in a binomial experiment with 20 trials and a probability of 0.45 on a single trial using the normal distribution, we need to calculate the z-score and find the corresponding probability from the standard normal distribution table.
First, we calculate the mean (μ) and standard deviation (σ) of the binomial distribution:
μ = n * p = 20 * 0.45 = 9
σ = sqrt(n * p * (1 - p)) = sqrt(20 * 0.45 * (1 - 0.45)) = 2.494
Next, we calculate the z-score using the formula:
z = (x - μ) / σ
where x represents the number of successes we are interested in (10 in this case).
z = (10 - 9) / 2.494 = 0.4016
Next, we use the standard normal distribution table or a calculator to find the probability corresponding to the z-score.
From the table, the probability for a z-score of 0.40 is approximately 0.6554.
Therefore, the probability of exactly 10 successes in the binomial experiment is approximately 0.6554 (rounded to the thousandths place).
To find the probability of exactly 10 successes in a binomial experiment with 20 trials and a probability of success of 0.45 on a single trial, we can use the normal distribution as an approximation.
First, we need to check if the conditions for using the normal approximation are satisfied. According to the normal approximation rule of thumb, the binomial distribution can be approximated by a normal distribution if n*p >= 10 and n*(1-p) >= 10. In this case, n = 20 and p = 0.45.
Checking the conditions:
n * p = 20 * 0.45 = 9
n * (1-p) = 20 * (1-0.45) = 11
Since both n * p and n * (1-p) are greater than 10, we can proceed with using the normal approximation.
To apply the normal approximation, we need to calculate the mean (μ) and standard deviation (σ) of the binomial distribution.
For a binomial distribution, μ = n * p and σ = √(n * p * (1-p)).
In this case, μ = 20 * 0.45 = 9 and σ = √(20 * 0.45 * (1-0.45)) = √(20 * 0.45 * 0.55) = 2.2136.
Now, we can calculate the z-score for exactly 10 successes:
z = (X - μ) / σ = (10 - 9) / 2.2136 = 0.4512
Next, we need to find the corresponding cumulative probability using the standard normal distribution table or a calculator.
Using a standard normal distribution calculator or table, we can find that the cumulative probability for a z-score of 0.4512 is 0.6741.
Therefore, the probability of exactly 10 successes in this binomial experiment using the normal distribution approximation is approximately 0.674.
Rounded to the thousandths place, the answer is 0.674.