The Mean Is the Sum of All Values Divided by the Number of Values

1. The Mean, Median and Standard Deviation

The mean is the sum of all values divided by the number of values:

475.62/30 = 15.854 ounces

The median value is the middle number in a sample/population. Since there is an even number of values in our sample (30) we can take the average of the 15th and 16th value to get our median value. In this case, the sample was sorter in order from least to greatest:

The median value = (15.98 + 16)/2 = 15.99 ounces

The standard deviation is the variability of the data itself. The sample standard deviation formula is calculated by the following equation:

Where x is the observed value, is the sample mean, and n is the sample size.

=0.6614 ounces

2. Construct a 95% Confidence Interval for the ounces in the bottles.

A 95% confidence interval is derived from the following equation:

Where = 15.854, t = 1.96 (the z-score required for a 95% level of confidence) and n = 30 (sample size). We get the following result:

15.854 .358

Therefore, we can be 95% confident that the range of all the bottles were filled with anywhere between 15.496 and 16.212 ounces of soda.

3. Conduct a hypothesis test to verify if the claim that a bottles contain less than sixteen (16) ounces is supported. Clearly state the logic of your test, the calculations, and the conclusion of your test.

To prove that there is evidence suggesting that the bottles are being filled less than what they should be, we can perform a 1-sample t-test to prove whether or not the complaints are valid. We start with the null hypothesis that the bottles are really being filled with a true 16 ounces. However since the average of the sample taken are slightly lower than 16 (15.854), we would draw up our alternative hypothesis that the bottles are being filled with a significant amount less than 16 ounces. Below entails the thought process:

Test: T-Test

Satisfied by the Central Limit Theorem (normal distribution of data at n)

Relevant Statistics:

n = 30, s = .6614,

Significance Level: .05

Results:

t = -1.2091, p = 0.1182 ∴ Reject

The p-value is the probability of committing a Type-1 Error or a false positive. There is an 11.82% chance of that being the case and at a 5% significance level, there is significant evidence that would suggest that the bottles are being underfilled.

One possible cause to this problem could be that the calibration for dispensing the soda into the bottles is off by about 1% from the average reading of our sample of 15.854. Another cause to the problem could lie in that the bottling plant counts the weight of the bottle itself in addition to the soda for the scales to reach 16 ounces. The bottle and its contents may weigh 16 ounces, but there is not 16 ounces of liquid and that is what is giving the low sample readings. A third possibility is that the scales themselves are faulty. There may be 16 ounces of liquid going into the container but the scale that measured the sample is incorrect.

The best way to solve for all three issues would be to have a control scale with an empty bottle and tare the scale. Then calculate how much pressure and time is needed to fill up the bottles with 16 ounces of soda exactly and then recalibrate all the machines for that setting and then the problem should be solved.