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Bottling Stats
The mean, median and standard deviation of this data set are as follows:
Mean
Median
STD Dev
The formula for the confidence interval is anything under the curve besides the top 2.5% and the bottom 2.5%.
So 1.96 * .5503 / (5.477) = 0.1969 is the margin of error.
The confidence interval therefore is 14.87 ± 0.1969, meaning that the lower bound is 14.67 and the upper bound is 15.07.
The question is asked wrong. If you want to test if a single bottle has 16 ounces or not, you measure the liquid. There are 30 different samples, each having performed this test. It is assumed that the question intends to ask about the average bottle, maybe, something like that.
The null hypothesis is that there is less than 16 ounces of soda in a typical bottle. The alternative hypothesis is that there are 16 ounces or more in a typical bottle.
To test this, you run the numbers of the 30 samples. The 95% confidence interval is 16 ± .1969, so 15.80 to 16.197. The actual population mean is 14.87, which is outside of the confidence interval.
The null hypothesis is therefore rejected -- the bottles contain a mean fill that is lower than the fill allowable within the 95% confidence interval. The complaint is correct -- the bottles are not sufficiently filled.
There are a couple of reasons why the bottles are not being filled properly. Most likely this is a calibration issue with the filling machine. The machine is chronically underfilling, which is most likely explained by calibration. The actual volatility of the fills (standard deviation) is high but does not explain the strong deviation from the expected mean of 16.0.
To solve this problem, I would check the calibration of the filler. Normally, the filler will fill to a level around the mean, usually at the 95% confidence interval -- or better if it is a more expensive filler. So if the machine is doing that here, it clearly thinks that 14.9 ounces is 16 ounces, or somebody changed the target fill level to 14.9 from 16. So the machine either needs to have its fill level target reset or it needs to be recalibrated in order to remedy this problem.
References
Yale. (1998). Confidence intervals. Yale.edu. Retrieved June 10, 2014 from http://www.stat.yale.edu/Courses/1997-98/101/confint.htm
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