## Mean, Median, and Standard Deviation

The first procedure of examining the soda amounts contained in bottles would require calculating the mean deviation. The mean deviation for ounces in the bottles would be calculated by summing every ounce measurement for every bottle and then dividing the sum by the number of bottles examined (which is thirty). Adding up the measured amount of soda contained in each of the sample bottles gives a result of 467.12 ounces. This sum divided by the number of bottles (30) results in 15.57 ounces. This is the determined mean deviation.

The problem with calculating mean deviation, however, is that the typical outcome for each of the measured samples. There may be an outcome that highly differs from the rest of the data, which will result in mean deviating becoming much less precise. The median deviation is an alternative that provides a more accurate deviation. Basically, an even number of outcomes presents an opportunity to take two middle outcomes and calculate the median deviation in the same fashion as the mean. The two middle outcomes for bottle samples are 15.88 and 15.37 ounces. Dividing those by two (which is the number of selected outcomes) equals 15.63 ounces. This is the determined median deviation.

Standard deviation is designed to demonstrate how the measured data is spread in both directions (minimum and maximum). To calculate standard deviation, one would first need to calculate the mean which is already done. The next step would require subtracting the mean from each of the selected outcomes (ounces of soda in bottles). The according results must then be separately squared and summed. Next, the result is divided by the number of outcomes minus one. This determines the variance. Standard deviation is calculated by extracting the square root of the variance. After the standard variation is determined, it must be separately added to the mean and subtracted from it. This will give us both the lowest and highest extremities of measurements. Calculations resulted in a standard deviation of 0.3. Thus, the minimum extremity is 15.57 – 0.3 = 15,27 and the maximum extremity is 15.57 + 0.3 = 15,87.

## Confidence Interval and Hypothesis Test

The confidence interval is where the true value of the measurements is detected. Basically, the confidence interval is represented by a range of values. To calculate a confidence interval for soda amounts in sample bottles, the following formula is applied:

Where x̄ is the mean deviation (15.57), Z is the constant value chosen for a 95% confidence interval (1.960), s is the standard deviation (0.3), and n is the total sample number (30). Applying this formula results in x̄ = 15.57 ± 0.21. This means that the true value lies somewhere between 15.36 and 15.78.

Thus, the most significant hypothesis test is already conducted by performing the examination of thirty sample bottles. Therefore, further testing may be viewed as excessive. This test demonstrated that both deviations and confidence intervals indicate that the average amount of soda in a bottle is indeed less than sixteen ounces. The only other way to detect insufficient soda amounts in bottles would be to examine every bottle produced rigorously. However, this would have nothing to do with statistical methods. This method of quality control would ensure that every bottle contains a sufficient amount of soda, although, if the shortage would be detected again, the reasons would still be unclear.

## Discussion

The reasons as to why the average amount of soda in bottles is less than sixteen ounces may vary considerably. However, they are most likely related to manufacturing. The first possible reason would then be that there are some malfunctions in the used equipment. This may be eliminated by performing a thorough diagnostics of every piece of equipment used in the process of distributing soda into bottles. Each piece must be examined and if the malfunctions are confirmed, repaired immediately.

Another possible reason may lie in the fact that some of the materials used in the manufacturing process lack quality. This may be the least probable of the indicated reasons. However, since there is a possibility, it must not be simply set aside. Thus, the materials used in manufacturing must be examined by experts to determine whether or not they are outdated or of low quality. The low quality of components, in turn, may be a result of the provider’s lack of professionalism. It may turn out that the provider (or providers) sold the company cheap or outdated materials to achieve higher benefits for themselves. If that is the case, the provider may be sued by the company’s leadership in accordance with the laws regulating trade and entrepreneur relationships.

Finally, the reason may be in the fact that some of the members of the company’s leadership are trying to gain some personal benefits by decreasing money spent on resources or maintenance of equipment. It may be reasonable to suspect this since there is a lot of possibilities to hide such small transgressions in the framework of such type of business. If that is indeed the case, the leadership must be thoroughly inspected. Every piece of documentation that covers the expenses of purchasing materials or maintaining equipment is to be examined. Naturally, if any fraud took place, it will be noticed. The company may then require reconstruction on various levels.