Final answer:
To approximate the binomial distribution with a normal distribution, we check if np and nq are both greater than 5. The probability of having at most 31 successes can be found by calculating the z-score using the formula z = (x - μ) / σ. Using a standard normal distribution table or calculator, we find the cumulative probability corresponding to the z-score.
Step-by-step explanation:
To approximate the binomial distribution with a normal distribution, we need to ensure that the quantities np and nq are both greater than 5. For this problem, n = 90 and p = 0.30, so np = 90 * 0.30 = 27 and nq = 90 * (1 - 0.30) = 63. Since both np and nq are greater than 5, we can use the normal distribution approximation.
To find the probability of having at most 31 successes, we need to find the cumulative probability up to 31. Using the normal distribution, we can find the z-score corresponding to 31 successes and then use a standard normal distribution table or calculator to find the cumulative probability. The formula for the z-score is z = (x - μ) / σ, where x is the number of successes, μ is the mean (np), and σ is the standard deviation (√(npq)).
Calculating the z-score: z = (31 - 27) / √(90 * 0.30 * (1 - 0.30)) = 4 / 4.743 = 0.843.
Using a standard normal distribution table or calculator, we find that the cumulative probability corresponding to a z-score of 0.843 is approximately 0.8008.