Step 1 :Calculate the standard error (SE) of the sample mean using the formula \(SE = \frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 39.8\) and \(n = 107\), we get \(SE = \frac{39.8}{\sqrt{107}} = 3.85\).
Step 2 :Calculate the z-score using the formula \(Z = \frac{X - \mu}{SE}\). Given \(X = 207\), \(\mu = 199\), and \(SE = 3.85\), we get \(Z = \frac{207 - 199}{3.85} = 2.0779\).
Step 3 :Look up the z-score in the standard normal distribution table or use a calculator to find the probability. The table or calculator gives the probability that the value is less than 207, so we need to subtract this from 1 to get the probability that the value is greater than 207. We get \(P(X > 207) = 1 - P(X < 207) = 1 - 0.9808 = 0.0192\).
Step 4 :\(\boxed{P(X > 207) = 0.0192}\)