To calculate the probability that daily production is between 21.5 and 40.3 liters for a normally distributed population, we can use the cumulative distribution function (CDF) of the normal distribution.
Given:
- Mean (μ) = 30 liters
- Standard deviation (σ) = 5.7 liters
We'll first convert the values 21.5 and 40.3 liters to Z-scores using the formula:
\[ Z = \frac{X - \mu}{\sigma} \]
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Calculate Z for 21.5 liters: \[ Z_{21.5} = \frac{21.5 - 30}{5.7} \approx \frac{-8.5}{5.7} \approx -1.4912 \]
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Calculate Z for 40.3 liters: \[ Z_{40.3} = \frac{40.3 - 30}{5.7} \approx \frac{10.3}{5.7} \approx 1.8035 \]
Next, we use the standard normal cumulative distribution function (Φ) to find the probabilities associated with these Z-scores.
Using a calculator or statistical software, we can find:
\[ P(Z < -1.4912) \quad \text{and} \quad P(Z < 1.8035) \]
Let’s find these probabilities. Using Python, R, or a statistical calculator, you would find:
- \(P(Z < -1.4912) \approx 0.0684\)
- \(P(Z < 1.8035) \approx 0.9644\)
Now, we calculate the probability that daily production is between 21.5 and 40.3 liters:
\[ P(21.5 < X < 40.3) = P(Z < 1.8035) - P(Z < -1.4912) \] \[ P(21.5 < X < 40.3) \approx 0.9644 - 0.0684 = 0.8960 \]
Therefore, the probability that the daily production is between 21.5 and 40.3 liters is approximately 0.8960.
So the final answer is:
Answer = 0.8960 (rounded to 4 decimal places).
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