Question 2 of 7. Step 1 of 1
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A soft drink machine outputs a mean of 29 ounces per cup. The machine's output is normally distributed with a standard deviation of 4 ounces. What is the probability of putting less than 31 ounces in a cup? Round your answer to four decimal places.

Answer

Step 1 ：The problem is asking for the probability of a normally distributed random variable being less than a certain value. This is a standard problem in statistics and can be solved using the Z-score formula and a Z-table or a function that calculates the cumulative distribution function (CDF) for a normal distribution.

Step 2 ：The Z-score formula is: $Z=\frac{X-\mu }{\sigma }$ where $X$ is the value we are interested in, $\mu$ is the mean, and $\sigma$ is the standard deviation. In this case, $X=31$, $\mu =29$, and $\sigma =4$.

Step 3 ：After calculating the Z-score, we can use a Z-table or a function that calculates the CDF for a normal distribution to find the probability.

Step 4 ：Substituting the given values into the Z-score formula, we get $Z=0.5$.

Step 5 ：Using a Z-table or a function that calculates the CDF for a normal distribution, we find that the probability corresponding to $Z=0.5$ is approximately 0.6915.

Step 6 ：Final Answer: The probability of the machine outputting less than 31 ounces is $\boxed{{\displaystyle 0.6915}}$.