Question 2 of 7. Step 1 of 1 $1 / 7$ Correct 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
Solution
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 \(\boxed{0.6915}\).
