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Bot GPT 4-o mini To find the probability that the length of a randomly selected steel rod is less than 200.7 cm, we first need to standardize this value by converting it into a z-score. The z-score is calculated using the formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) is the value we're interested in (200.7 cm),
- \(\mu\) is the mean (201.9 cm),
- \(\sigma\) is the standard deviation (0.9 cm).
Plugging in the values, we get:
\[
z = \frac{(200.7 - 201.9)}{0.9} = \frac{-1.2}{0.9} \approx -1.3333
\]
Next, we need to find the probability corresponding to this z-score. We can look up the z-score in a standard normal distribution table or use a calculator or software that provides the cumulative distribution function (CDF) for the standard normal distribution.
Using a standard normal distribution table or calculator, we get:
\[
P(Z < -1.3333) \approx 0.0918
\]
Thus, the probability that the length of a randomly selected steel rod is less than 200.7 cm is:
\[
P(X < 200.7) \approx 0.0918
\]
Therefore, the final answer, accurate to four decimal places, is:
\[
\boxed{0.0918}
\]