To find the probability that the length of a randomly selected steel rod is greater than 162 cm, we first need to calculate the z-score for 162 cm using the formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:
- \(X\) is the value we are interested in (162 cm),
- \(\mu\) is the mean (164.4 cm), and
- \(\sigma\) is the standard deviation (1.4 cm).
Substituting the values into the formula gives us:
\[ z = \frac{(162 - 164.4)}{1.4} = \frac{-2.4}{1.4} \approx -1.7143 \]
Next, we need to find the probability \(P(X > 162)\), which is equivalent to finding \(P(Z > -1.7143)\). Using the standard normal distribution table or a calculator, we first find \(P(Z \leq -1.7143)\).
Using a standard normal distribution calculator or table, we get:
\[ P(Z \leq -1.7143) \approx 0.0436 \]
Thus, the probability \(P(Z > -1.7143)\) is:
\[ P(Z > -1.7143) = 1 - P(Z \leq -1.7143) \approx 1 - 0.0436 = 0.9564 \]
Therefore, the probability that the length of a randomly selected steel rod is greater than 162 cm is:
\[ P(X > 162 \text{ cm}) \approx 0.9564 \]
Final answer:
\[ \boxed{0.9564} \]