statistics

Question

he amount of coffee that a filing machine puts into an 8 ounce jar is normally distributed with a mean of 8.2 ounces and a standard deviation of 0.18 ounce what is the probability that the sampling error made in the estimating amount of coffee for all 8 ounces jars by the mean by the mean of a random sample of 100 jars will be at most?

Answer 1

Our random variable of interest is the amount of coffee poured into an 8oz jar. Let's call that X and we know that it's Normally distributed with a mean of 8.2 oz and a standard deviation of 0.18 oz.

But the question isn't looking at any one jar -- the question is asking about the sampling error when we take a random sample of 100 8oz jars. The sampling error (SE) may be calculated as the standard deviation divided by the square root of the sample size. I can't write that easily in this format but it basically looks like σ/sqrt(n). We can use this to set up a formula as:

SE = 0.18/sqrt(100)

Or more simply

SE = 0.18/10 = 0.018.

I think that there is something missing from the posted question that allows me to calculate any kind of probabilities for the sample mean but this is the step we need to be able to make them work. Let's say that the question asks for the probability that the sample mean of a random sample of 100 8oz jars is bigger than 8.195 oz. The sampling distribution of the sample mean is Normal with the same mean as the population (8.2 oz) but the new standard deviation is the SE (0.018 oz). From here it's just like any other normal distribution problem with the final answer being 0.6094

If there are more specifics to the question please post them and I can give better answers.