To find the probability that the daily production of a herd of cows is between 21.5 and 40.3 liters, we'll use the properties of the normal distribution. Specifically, we will standardize the values to find their corresponding Z-scores and then utilize the cumulative distribution function (CDF) for the standard normal distribution.
Step 1: Calculate the Z-scores.
The formula to convert a raw score \(X\) to a Z-score is:
\[ Z = \frac{X - \mu}{\sigma} \]
Where:
- \(X\) is the raw score,
- \(\mu\) is the mean (30 liters),
- \(\sigma\) is the standard deviation (5.7 liters).
For \(X = 21.5\) liters:
\[ Z_{21.5} = \frac{21.5 - 30}{5.7} = \frac{-8.5}{5.7} \approx -1.4912 \]
For \(X = 40.3\) liters:
\[ Z_{40.3} = \frac{40.3 - 30}{5.7} = \frac{10.3}{5.7} \approx 1.8035 \]
Step 2: Find probabilities using the CDF.
To find the probability of being between these two Z-scores, we will look for the cumulative probabilities for \(Z_{21.5}\) and \(Z_{40.3}\).
We can use a computational tool (like Python, R, or a scientific calculator) to find the CDF values directly for the Z-scores.
Using the cumulative distribution function \( \Phi(z) \):
- For \(Z = -1.4912\): \( \Phi(-1.4912) \)
- For \(Z = 1.8035\): \( \Phi(1.8035) \)
Using a standard normal distribution calculator or a statistical software:
\[ \Phi(-1.4912) \approx 0.0685 \] \[ \Phi(1.8035) \approx 0.9640 \]
Step 3: Calculate the probability between the two Z-scores.
The probability \(P(21.5 < X < 40.3)\) is given by:
\[ P(21.5 < X < 40.3) = \Phi(1.8035) - \Phi(-1.4912) \]
Substituting the values:
\[ P(21.5 < X < 40.3) \approx 0.9640 - 0.0685 = 0.8955 \]
Final Answer (rounded to 4 decimal places):
\[ \text{Probability} = 0.8955 \]
So the final answer is:
Answer = 0.8955