To find the probability that the length of a randomly selected steel rod is between 190 cm and 198.8 cm, we will first calculate the z-scores for both lengths using the formula:
\[ z = \frac{X - \mu}{\sigma} \]
where:
- \( \mu = 195.9 \) cm (mean),
- \( \sigma = 2.1 \) cm (standard deviation),
- \( X \) is the value for which we want to calculate the z-score.
Step 1: Calculate the z-scores
For \( X = 190 \) cm:
\[ z_1 = \frac{190 - 195.9}{2.1} = \frac{-5.9}{2.1} \approx -2.8095 \]
For \( X = 198.8 \) cm:
\[ z_2 = \frac{198.8 - 195.9}{2.1} = \frac{2.9}{2.1} \approx 1.3809 \]
Step 2: Find the probabilities using the z-scores
Now, we will use the standard normal distribution table (or a calculator) to find the probabilities corresponding to these z-scores.
Probability for \( z_1 \approx -2.8095 \):
Using a standard normal distribution calculator or table, we find:
\[ P(Z < -2.8095) \approx 0.0025 \text{ (this may vary slightly depending on the resource used)} \]
Probability for \( z_2 \approx 1.3809 \):
Again using a standard normal distribution calculator or table, we find:
\[ P(Z < 1.3809) \approx 0.9162 \text{ (this may vary slightly depending on the resource used)} \]
Step 3: Find the probability between the two z-scores
Now we can find the probability that a randomly selected rod is between 190 cm and 198.8 cm by subtracting the probabilities:
\[ P(190 < X < 198.8) = P(Z < 1.3809) - P(Z < -2.8095) \]
Substituting the values we found:
\[ P(190 < X < 198.8) \approx 0.9162 - 0.0025 = 0.9137 \]
Thus, the probability that the length of a randomly selected steel rod is between 190 cm and 198.8 cm is approximately:
\[ \boxed{0.9137} \]
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