Answer :
Final answer:
The probability that the mean of a sample of sixty-four 16-ounce bottles of water is less than 15.7 ounces is approximately 0.38%, calculated using the principles of sampling distributions and z-scores in probability and statistics.
Explanation:
This question falls under the field of Probability and Statistics, specifically the concept of Sampling Distributions. To solve this, you need to first understand some basic concepts. The problem tells us the mean (15.9 ounces) and standard deviation (0.6 ounces) of the population.
When we draw a sample, we expect the mean to be close to the population mean, but there is some variation. This variation is quantified by the standard error, which is the standard deviation divided by the square root of the sample size. In this case, the standard error is 0.6 ounces / sqrt(64) = 0.075 ounces.
We're interested in the probability that the sample mean is less than 15.7 ounces. To find this, we need to standardize 15.7 using the z-score formula. To find the z-value, we subtract the mean and divide by the standard error. So, Z = (15.7 - 15.9) / 0.075 = -2.67.
Using a z-table, we find that the probability of getting a z-score less than -2.67 is approximately 0.0038 or 0.38%. So, there is a 0.38% chance that the mean of a sample of sixty-four 16-ounce bottles of water is less than 15.7 ounces.
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