Assume the data set described is normally distributed with the given mean and standard deviation, and with $n$ total values. Find the approximate number of data values that will fall in the given range.
\[
M e a n=440
\]
Standard deviation $=14$
\[
n=500
\]
Range: 398 to 482
In this case, we expect about data values to fall between 398 and 482 .

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 ：\(\boxed{\text{Approximately 499 data values will fall between 398 and 482}}\)