To find the probability that a randomly selected plant is between 20.2 cm and 24.9 cm tall, we can use the properties of the normal distribution, based on the information given.
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Calculate the Z-scores for the heights of 20.2 cm and 24.9 cm.
The formula for calculating the Z-score is:
\[ Z = \frac{(X - \mu)}{\sigma} \]
where:
- \(X\) is the value for which we are finding the Z-score,
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.
Given:
- \(\mu = 20\) cm,
- \(\sigma = 3\) cm.
For \(X = 20.2\) cm:
\[ Z_{20.2} = \frac{(20.2 - 20)}{3} = \frac{0.2}{3} \approx 0.0667 \]
For \(X = 24.9\) cm:
\[ Z_{24.9} = \frac{(24.9 - 20)}{3} = \frac{4.9}{3} \approx 1.6333 \]
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Use a Z-table or a calculator to find the probabilities corresponding to these Z-scores.
- For \(Z_{20.2} \approx 0.0667\), the corresponding probability (from the Z-table or calculator) is approximately \(0.5279\).
- For \(Z_{24.9} \approx 1.6333\), the corresponding probability is approximately \(0.9484\).
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Calculate the probability that a randomly selected plant is between these two heights:
\[ P(20.2 < X < 24.9) = P(Z < 1.6333) - P(Z < 0.0667) \]
\[ P(20.2 < X < 24.9) \approx 0.9484 - 0.5279 = 0.4205 \]
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Convert the probability to a percentage:
\[ 0.4205 \times 100 \approx 42.05% \]
So, the probability that a randomly selected plant is between 20.2 cm and 24.9 cm tall is 42.05%.