Questions: A soda machine outputs a mean of 23 ounces per cup. The machine's output is normally distributed with a standard deviation of 4 ounces. What is the probability of filling a cup between 27 and 31 ounces?

A soda machine outputs a mean of 23 ounces per cup. The machine's output is normally distributed with a standard deviation of 4 ounces. What is the probability of filling a cup between 27 and 31 ounces?
Transcript text: A soda machine outputs a mean of 23 ounces per cup. The machine's output is normally distributed with a standard deviation of 4 ounces. What is the probability of filling a cup between 27 and 31 ounces?
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Solution

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Solution Steps

Step 1: Calculate the CDF at \( x_1 = 27 \)

To find the cumulative distribution function (CDF) at \( x_1 = 27 \), we use the formula for the CDF of a normal distribution:

\[ P(X \leq 27) = CDF(27) = 0.8413 \]

Step 2: Calculate the CDF at \( x_2 = 31 \)

Next, we calculate the CDF at \( x_2 = 31 \):

\[ P(X \leq 31) = CDF(31) = 0.9772 \]

Step 3: Determine the Probability Between \( x_1 \) and \( x_2 \)

The probability of filling a cup between 27 and 31 ounces is given by the difference of the two CDF values:

\[ P(27 < X < 31) = P(X \leq 31) - P(X \leq 27) = 0.9772 - 0.8413 = 0.1359 \]

Final Answer

\(\boxed{0.1359}\)

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