Question 1172102: Assume the fill amount of bottles of a soft drink is normally distributed with a mean of 2.0 liters and standard deviation of 0.04 liter. If we take random samples of 12 bottles, Find the sample mean fill amount that is lower than 95% of all such sample means.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! population mean = 2 liters.
population standard deviation = .04 liters.
sample size is 12.
standard error = population standard deviation / sqrt(sample size) = .04 / sqrt(12) = .011547
you are looking for a sample mean that is lower than 95% of all such sample means.
i think this means that, if you took an infinite number of samples of size 12 and looked at all the means of those samples, your sample mean would need to be lower than 95% of those sample means.
this looks like it might have something to do with confidence levels.
at 95% confidence level, the one tailed alpha would be .05.
the critical z-score would be -1.64485.
the z-score formula is z = (x - m) / s
z is the z-score
x is the raw score
m is the raw mean
s is the standard error
in your problem, this becomes:
-1.64485 = (x - 2) / .011547
solve for x to get:
x = -1.64485 * .011547 + 2 = 1.981006917.
i believe that means that a sample mean of less than 1.981006917 will be less than 95% of all sample means for samples of size 12 where the population mean is 2 and the population standard deviation is .04.
keep in mind that, when you are dealing with sample means, you use the standard error and not the standard deviation.
the standard error is the standard deviation of the distribution of sample means.
if you take 100 samples of a certain size, you will have 100 sample means and, if you find the standard deviation of that distribution of sample means, you will have your standard error.
the standard distribution of sample means, otherwise known as the standard error, is affected by the sample size.
the larger the sample size, the smaller the standard deviation of the distribution of sample means.
this is because the distribution of sample means will be closer to the population mean, hence less variability between samples, when the sample size is larger.
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