Answer:
To solve this problem, we need to use the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
First, we need to standardize the length of the rod of interest (19.9 cm) using the formula:
z = (x - mu) / sigma
where z is the standardized score, x is the length of the rod, mu is the mean length of the rods (20 cm), and sigma is the standard deviation of the lengths of the rods (0.8 cm).
Substituting the given values, we get:
z = (19.9 - 20) / 0.8
z = -0.125
Next, we need to find the proportion of rods that have a length less than 19.9 cm, which is equivalent to finding the area to the left of z = -0.125 on the standard normal distribution curve.
We can use a standard normal distribution table or a calculator to find this area. Using a standard normal distribution table, we can look up the area corresponding to z = -0.125, which is approximately 0.4502.
Therefore, the proportion of rods that have a length less than 19.9 cm is approximately 0.4502 or 45.02%.