Patty's pastries has 400 employees. It has been estimated that the probability of a worker gaining 6 pounds per year is .2. Use the normal approximation to the binomial random variable to find the probability that at least 90 of her workers will gain 6 pounds this year.

I don't know how to begin this problem... n=400, p=.2, q=1-.2=.8, p (x=90)?

To begin solving this problem using the normal approximation to the binomial random variable, there are a few steps you need to follow:

Step 1: Find the mean and standard deviation of the binomial distribution.

The mean of a binomial distribution is μ = n * p, where n is the number of trials and p is the probability of success. In this case, n = 400 and p = 0.2, so the mean is μ = 400 * 0.2 = 80.

The standard deviation of a binomial distribution is σ = sqrt(n * p * q), where q is the probability of failure (1 - p). In this case, q = 1 - 0.2 = 0.8. So, the standard deviation is σ = sqrt(400 * 0.2 * 0.8) = 8.

Step 2: Convert the problem into an equivalent normal distribution problem.

To use the normal distribution, you need to calculate the probability of at least 90 workers gaining 6 pounds in a year. This can be written as P(X >= 90), where X is the binomial random variable.

Step 3: Calculate the z-score.

The z-score is a measure of how many standard deviations a particular value is away from the mean. The formula to calculate the z-score is z = (x - μ) / σ, where x is the desired value (90 in this case), μ is the mean, and σ is the standard deviation. So, we have z = (90 - 80) / 8 = 1.25.

Step 4: Find the probability using the z-score.

Use a standard normal distribution table or a calculator to find the probability corresponding to the z-score of 1.25. The probability is the area under the curve to the right of the z-score. Since we're looking for the probability of at least 90 workers gaining 6 pounds, we need to find the probability of X being greater than or equal to 90.

Step 5: Calculate the final probability.

The final probability is the probability of X being greater than or equal to 90, which can be obtained from the previous step.

So, by following these steps, you'll be able to find the probability that at least 90 of Patty's workers will gain 6 pounds this year using the normal approximation to the binomial random variable.