Introduction
As the manager of a bottling company whose customers have launched complaints about the amount of soda that is less than recommended 16 ounces, I have undertaken investigations to ascertain the problem and report to my boss. To get valid and reliable data, I directed employees to sample 30 bottles randomly during different shifts. Employees then measured the amount of soda in each of the 30 bottles and tabulated the findings for further analysis. To determine if the claim launched by customers is true, the case study will calculate descriptive statistics, construct a 95% confidence interval, perform a hypothesis test, discuss the causes of the claim, and offer an appropriate recommendation.
Main body
Descriptive statistics were calculated using MS Excel functions and tabulated in the table below.
Table 1. Descriptive Statistics.
The mean for ounces in the 30 bottles sampled (M = 15.954) is less than the recommended 16 ounces. The standard deviation (SD = 0.855) indicates that the amount of soda in each bottle is highly variable (15.954±0.855). Since the median (16.01) is greater than the mean (15.954), it implies the distribution of data has a negative skew.
The 95% confidence interval was constructed using mean (15.954), standard deviation (0.855), critical value (1.96), and sample size (n = 30).
Standard error = standard deviation/√sample size = 0.855/√30 = 0.1561
Margin of error = Critical value (Z) x standard error = 1.96 x 0.1561 = 0.30596
95% Confidence Interval = Mean ±Margin of Error = 15.954±0.30596 (15.648, 16.260)
Therefore, the lower limit and the upper limit of 95% confidence interval of a true mean of the amount of soda that the company produces are 15.648 and 16.260 ounces respectively. The confidence level shows that the true mean of the population could be less than 16 ounces.
The study performed a hypothesis test to verify or refute the claim that the amount of soda in bottles is less than 16 ounces, The one-sample t-test is an appropriate statistical test because the study entails a comparison of sample and hypothesized means for ounces of soda in bottles.
The case analysis entailed the formulation of the following hypotheses:
Null hypothesis: The mean amount of soda in bottles is equaled 16 ounces.
μ = 16
Alternative hypothesis: The mean amount of soda in bottles is less than 16 ounces.
µ < 16
Table 2 shows that the one-sample t-test fails to reject the null hypothesis that the mean amount of soda in bottles is 16 ounces, t(29) = -2948, p = 0.3851. The logic of the hypothesis test is that the t-value is not in the rejection region and the p-value is greater than the significance level of 0.05. Hence, the conclusion is that the claim of customers is not true since the mean of ounces does not differ significantly from the advertised value of 16 ounces.
Table 2. One-Sample T-Test.
Conclusion
Statistical analysis has disapproved the claim made by customers that the amount of soda is less than 16 ounces. The reason behind the claim is that a substantial proportion of bottles contain less than 16 ounces of soda. Since the average volume of soda is 15.954 ounces, it shows that a significant number of bottles contain less than the recommended amounts of 16 ounces. Moreover, variations in the mean (15.954±0.855) and the 95% confidence interval (15.954±0.30596) show that over half of bottles contain less than 16 ounces of soda. Thus, to mitigate the issue, the company ought to improve its quality control measures to ensure that the mean of bottled soda is within the recommended range, and the standard deviation and 95% confidence interval do not include values less than 16 ounces.