Question

Individual bottles of water are filled by a machine at a factory with an amount of water that is approximately normal with a mean of 505\,\text{mL}505mL505, start text, m, L, end text and a standard deviation of 10\,\text{mL}10mL10, start text, m, L, end text. A random sample of 161616 bottles is selected for a quality inspection. What is the probability that the mean amount of water in these 161616 bottles \bar x x ˉ x, with, \bar, on top is within 5\,\text{mL}5mL5, start text, m, L, end text of the population mean?

Answer 1

The probability that the mean amount of water in the 16 bottles is within 5 mL of the population mean is 50%.

Step-by-step explanation:

To find the probability that the mean amount of water in the 16 bottles is within 5 mL of the population mean, we need to calculate the z-score for the given values. The z-score formula is z = (x - µ) / (σ / sqrt(n)), where x is the sample mean, µ is the population mean, σ is the population standard deviation, and n is the sample size.

Plugging in the values, we have z = (505 - 505) / (10 / sqrt(16)) = 0. The z-score of 0 represents the mean of the sample, which is equal to the population mean. Therefore, the probability is 50%, as half the values lie below the mean in a normal distribution.

Answer 2

P(500< xˉ <510)≈0.95

Step-by-step explanation:

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