Final answer:
To find the probability that the sample mean will exceed 200, we need to calculate the z-score and use the normal distribution. The probability is approximately 0.0418, or 4.18%.
Step-by-step explanation:
To find the probability that the sample mean will exceed 200, we need to calculate the z-score and use the normal distribution. First, let's calculate the standard error of the sample mean using the population standard deviation and the finite population correction factor. The formula for the standard error is:
SE = sqrt((σ^2)*(1 - n/N)) / sqrt(n)
Where σ is the population standard deviation, n is the sample size, and N is the population size. Plugging in the values from the question, we get:
SE = sqrt((20^2)*(1 - 100/350)) / sqrt(100) = 2.8944
Next, we calculate the z-score using the formula:
z = (x - μ) / SE
Where x is the sample mean, μ is the population mean, and SE is the standard error. Plugging in the values from the question, we get:
z = (200 - 195) / 2.8944 = 1.7279
Finally, we can look up the probability of the z-score exceeding 1.7279 in the standard normal distribution table or use a calculator. The probability is approximately 0.0418, or 4.18%.