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------------------------------------------------ Bottles filled by a certain machine are supposed to contain 12 oz of liquid. In fact, the fill volume is random with a mean of 12.01 oz and a standard deviation of 0.20 oz. What is the probability that the mean volume of a random sample of 144 bottles is less than 12 oz?

The subject of this question is mathematics, specifically, statistics and probability. To determine the probability the mean volume of a random 144 bottles being less than 12 oz, you can use the Central Limit Theorem which states when the sample size (n) is large enough, the sampling distribution of the sample mean becomes approximately normal regardless of the shape of the population distribution.

First, we need to calculate the mean and standard deviation of the sampling distribution. The mean of this distribution, often called the expected value of the distribution, is equal to the mean of the population, which is 12.01 oz. The standard deviation of the distribution is the standard deviation of the population divided by the square root of the sample size (n), which is (0.20 oz / sqrt(144)) = 0.0167 oz.

The next step is to find the Z-score. The Z-score is a statistical measurement that represents the number of standard deviations a data point is from the mean. To find this, subtract the population mean from the sample mean, then divide by the standard deviation of the sampling distribution. So Z = (12 - 12.01) / 0.0167 = - 0.6. Looking up in a Z-table, we can find that the probability of having a value less than -0.6 is 0.2743, or 27.43%.

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