Answer :
Final answer:
The probability that the mean volume of a random sample of 169 bottles is less than 12 oz, given a population mean of 12.01 oz and a standard deviation of 0.2 oz, is approximately 0.2578, or 25.78%.
Explanation:
To find the probability that the mean volume of the bottles is less than 12 oz, we'll use the concept of the sampling distribution of the sample mean. Given that the mean volume (μ) is 12.01 oz and the standard deviation (σ) is 0.2 oz for a single bottle, the standard error (SE) for a sample of size 169 bottles can be calculated using the formula SE = σ / √n, where n is the sample size. Here, SE = 0.2 / √169 = 0.2 / 13 = 0.01538 oz.
Since the distribution of the sample mean is approximately normal (thanks to the Central Limit Theorem), we can use the standard normal distribution to find the probability. The Z-score for a sample mean of 12 oz can be calculated using Z = (X - μ) / SE, where X is the sample mean we're interested in (12 oz in this case). Thus, Z = (12 - 12.01) / 0.01538 = -0.65.
Using standard normal distribution tables, or a calculator, we find the probability corresponding to a Z-score of -0.65. This gives us a probability of approximately 0.2578. Hence, the probability that the mean volume of a random sample of 169 bottles is less than 12 oz is about 0.2578, or 25.78%.
