Based on historical weather data in a certain city, about 62% of the days are cloudy. Find the mean, standard deviation, and variance for the number of cloudy days in a 30-day month.

• A. Mean = 18.6 days; Standard deviation = 2.66 days; Variance = $7.08days^2$
• B. Mean = 18.6 days; Standard deviation = 3.38 days; Variance = $11.4days^2$
• C. Mean = 11.4 days; Standard deviation = 2.66 days; Variance = $7.08days^2$
• D. Mean = 11.4 days; Standard deviation = 3.38 days; Variance = $11.4days^2$

Answer: (A)

Verified step by step guidance

1. Identify the type of probability distribution: Since we are dealing with a fixed number of trials (30 days) and each day is either cloudy or not, this is a binomial distribution.
2. Calculate the mean of the binomial distribution: The mean (μ) is given by the formula μ = n * p, where n is the number of trials (30 days) and p is the probability of success (cloudy day, 0.62).
3. Calculate the variance of the binomial distribution: The variance (σ²) is given by the formula σ² = n * p * (1 - p), where n is the number of trials (30 days) and p is the probability of success (cloudy day, 0.62).
4. Calculate the standard deviation of the binomial distribution: The standard deviation (σ) is the square root of the variance, σ = √(σ²).
5. Interpret the results: The mean represents the expected number of cloudy days in a 30-day month, the variance measures the spread of the number of cloudy days, and the standard deviation provides a measure of the average deviation from the mean.