Step 1 :Given a normal distribution with mean \(\mu = 440\), standard deviation \(\sigma = 14\), and \(n = 500\) total data values.
Step 2 :Find the z-scores for the lower and upper bounds of the range (398, 482):
Step 3 :\(z_{lower} = \frac{398 - 440}{14} = -3.0\)
Step 4 :\(z_{upper} = \frac{482 - 440}{14} = 3.0\)
Step 5 :Use a z-table or cumulative distribution function (CDF) to find the probabilities corresponding to the z-scores:
Step 6 :\(P(z_{lower}) = 0.0013\)
Step 7 :\(P(z_{upper}) = 0.9987\)
Step 8 :Find the probability of the data falling within the range (398, 482):
Step 9 :\(P(398 \leq x \leq 482) = P(z_{upper}) - P(z_{lower}) = 0.9987 - 0.0013 = 0.9973\)
Step 10 :Approximate the number of data values that fall within the range (398, 482):
Step 11 :\(n_{range} = n \times P(398 \leq x \leq 482) = 500 \times 0.9973 = 498.65\)
Step 12 :\(\boxed{\text{Approximately 499 data values will fall between 398 and 482}}\)