Questions: A soft drink machine outputs a mean of 26 ounces per cup. The machine's output is normally distributed with a standard deviation of 3 ounces. What is the probability of filling a cup between 23 and 30 ounces? Round your answer to four decimal places.

A soft drink machine outputs a mean of 26 ounces per cup. The machine's output is normally distributed with a standard deviation of 3 ounces. What is the probability of filling a cup between 23 and 30 ounces? Round your answer to four decimal places.
Transcript text: A soft drink machine outputs a mean of 26 ounces per cup. The machine's output is normally distributed with a standard deviation of 3 ounces. What is the probability of filling a cup between 23 and 30 ounces? Round your answer to four decimal places.
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Solution

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Solution Steps

Step 1: Define the Parameters

The soft drink machine outputs a mean of \( \mu = 26 \) ounces per cup, with a standard deviation of \( \sigma = 3 \) ounces. The output is normally distributed.

Step 2: Calculate the CDF for 23 Ounces

To find the probability of filling a cup with less than or equal to 23 ounces, we calculate the cumulative distribution function (CDF) at \( x = 23 \): \[ P(X \leq 23) = CDF(23) \approx 0.1587 \]

Step 3: Calculate the CDF for 30 Ounces

Next, we calculate the CDF at \( x = 30 \) ounces to find the probability of filling a cup with less than or equal to 30 ounces: \[ P(X \leq 30) = CDF(30) \approx 0.9088 \]

Step 4: Determine the Probability Between 23 and 30 Ounces

The probability of filling a cup between 23 and 30 ounces is given by the difference of the two CDF values: \[ P(23 < X < 30) = P(X \leq 30) - P(X \leq 23) \approx 0.9088 - 0.1587 = 0.7501 \]

Step 5: Final Result

The probability of filling a cup between 23 and 30 ounces is approximately \( 0.7501 \).

Final Answer

\(\boxed{0.7501}\)

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