To find the probability that a value of X rounds to 84, we need to calculate the probability that X is between 83.5 and 84.5.

Standardize the values using the formula for the standard normal distribution:

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

where \(\mu = 82\) and \(\sigma = \sqrt{126}\).

Calculate the standardized values:

\(Z_1 = \frac{84.5 - 82}{\sqrt{126}} = 0.2227\)

\(Z_2 = \frac{83.5 - 82}{\sqrt{126}} = 0.1336\)

Find the probabilities using the standard normal distribution table:

\(\Phi(0.2227) = 0.5883\)

\(\Phi(0.1336) = 0.5533\)

Subtract the probabilities to find the probability that X rounds to 84:

\(P(83.5 < X < 84.5) = \Phi(0.2227) - \Phi(0.1336) = 0.5883 - 0.5533 = 0.0350\)