(i) We know that 25% of infections clear up in less than 7 days, which corresponds to a z-score of approximately -0.674. Using the standardization formula:

\(z = \frac{X - \mu}{\sigma}\)

Substitute the known values:

\(-0.674 = \frac{7 - \mu}{2.6}\)

Solving for \(\mu\):

\(-0.674 \times 2.6 = 7 - \mu\)

\(-1.7524 = 7 - \mu\)

\(\mu = 7 + 1.7524 = 8.75\)

(ii) We need to find \(P(X > 6.2)\) where \(X\) is normally distributed with mean 6.5 and standard deviation 2.6. Standardizing:

\(P(X > 6.2) = P\left( Z > \frac{6.2 - 6.5}{2.6} \right)\)

\(= P(Z > -0.1154)\)

Using the symmetry of the normal distribution, \(P(Z > -0.1154) = 1 - P(Z < -0.1154)\). Since \(P(Z < -0.1154) \approx 0.454\),

\(P(Z > -0.1154) = 1 - 0.454 = 0.546\)