# Bottling Company Case Study

Due to the frequent complaints about the bottle of soda containing less than advertised 16 ounces, I decided to check if these claims have grounds. Thirty bottles of soda were chosen randomly from different shifts and were weighed. The first approximation of the problem was to find the mean by summing all the weights and dividing their sum by the sample size. Having made simple calculations, I found that the average weight of one bottle was 14.87 ounces. The median and the standard deviation were 14.8 and 0.55 respectively.

To detect whether the sufficient cause to think that customers are right exists, I decided to find the 95% confidence interval about the mean. As the population standard deviation was unknown, I decided to use t-distributions. It is known that probability was 95%, α was 0.05, and α/2 was 0.025. Since the sample size was 30, there were 29 degrees of freedom. Using the table for t-distribution, I obtained 2.045 for t_{α/2}. The maximal error of an estimate was calculated with formula t_{α/2}(S/√n) and was equal to 0.2053. Therefore, I can assert with the 95% probability that the true value of average lies between 14.6647 and 15.0753.

After that, I put forward two hypotheses. The null hypothesis stated that the average amount of soda in bottle equals sixteen ounces. The alternative hypothesis claimed that its quantity is less than sixteen ounces. I chose the level of significance of 0.01.This fact meant that probability that I reject the null hypothesis when it is true was 1%. In order to define whether to reject the null hypothesis or not, I used the t-test since I did not know population standard deviation. Since the claim was µ< 16, I had to use the left-tailed t-test. The table for t-distribution gave me the critical region of – 2.462 and less. The test value was calculated with formula

Expected value was that indicated on the label. It was sixteen ounces. Observed value was the mean of the sample. The standard error was S/√n. Test value was found to be -11.95. This value lies within the critical region, and it means that the null hypothesis must be rejected. In fact, the test value is very far to the left. I think that t-tests would give the same results with much less probability of making type one error. I am sure that the problem of bottles having less than sixteen ounces exists and should be solved as soon as possible.

I suppose that there may be several reasons for the weight of soda in the bottle being less than sixteen ounces. On the one hand, the problem might be caused with the fact that the size of bottles is not up to standard. On the other hand, there may be malfunction of the filling system. One more reason to consider is that there is a considerable standard deviation for the weight of soda. Finally, trouble might origin from changing the density of soda. As a result, having the same value, it weighs less. It is necessary to check all the technological process to find maladjustment. In order to avoid such problems in future, I propose to carry out regular checks of the weight of soda. To avoid the misunderstanding on the part of customers, it seems advisable to indicate on the bottles not only the net weight but also the maximal possible error. In this way, they will know that the weight of soda may be not only smaller, but also higher than the value on the label.

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