Questions: A soft drink machine outputs a mean of 25 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 22 and 29 ounces? Round your answer to four decimal places.

A soft drink machine outputs a mean of 25 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 22 and 29 ounces? Round your answer to four decimal places.

Transcript text: A soft drink machine outputs a mean of 25 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 22 and 29 ounces? Round your answer to four decimal places.

Solution

Solution Steps

Step 1: Calculate the CDF at \( x_1 = 22 \) ounces

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

\[ P(X \leq 22) = CDF(22) = 0.2266 \]

Step 2: Calculate the CDF at \( x_2 = 29 \) ounces

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

\[ P(X \leq 29) = CDF(29) = 0.8413 \]

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

The probability of filling a cup between 22 and 29 ounces is given by the difference of the two CDF values:

\[ P(22 < X < 29) = P(X \leq 29) - P(X \leq 22) = 0.8413 - 0.2266 = 0.6147 \]

Final Answer

\(\boxed{0.6147}\)

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