An automatic machine that fills bags of unpopped popcorn is operating properly if the weights are independently and normally distributed with a mean of 114 grams and a standard deviation of 4.9 grams. Find the probability that if 7 bags are randomly selected, their mean weight exceeds 115 grams.
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To find the probability that the mean weight of 7 randomly selected bags exceeds 115 grams, we can use the Central Limit Theorem.
The Central Limit Theorem states that if you have a large enough sample size (in this case, 7 bags) and if the population distribution (weights of bags) is approximately normally distributed, then the distribution of the sample means will also be approximately normally distributed, regardless of the shape of the original population distribution.
The mean of the sample means (also known as the sample mean of the population) will be equal to the mean of the original population distribution, which is 114 grams.
The standard deviation of the sample means (also known as the standard error of the mean) will be equal to the standard deviation of the original population distribution divided by the square root of the sample size. In this case, the standard deviation of the original population distribution is 4.9 grams and the sample size is 7.
Thus, the standard error of the mean is:
standard deviation/original sample size = 4.9/√7 ≈ 1.85 grams
Now, we can convert our problem into a standard normal distribution problem by calculating the z-score.
The z-score is given by:
z = (x - μ) / σ ,
where x is the value we are interested in (115 grams), μ is the mean of the original population distribution (114 grams), and σ is the standard error of the mean (1.85 grams).
Now, let's calculate the z-score:
z = (115 - 114) / 1.85 ≈ 0.54
Finally, to find the probability that the mean weight of 7 bags exceeds 115 grams, we need to find the area under the standard normal distribution curve to the right of the calculated z-score.
Using a standard normal table or a calculator, we find that the probability corresponding to a z-score of 0.54 is approximately 0.2932. Therefore, the probability that the mean weight of 7 bags exceeds 115 grams is approximately 0.2932, or 29.32%.
Note: It's important to keep in mind that the Central Limit Theorem assumes independence of the samples.