Customers are complaining that the bottles of the brand soda produced in the firm contain less than the advertised 16 ounces of the product. The mean is defined as an average of observed outcomes (Heiman, 2013). In this context, observed outcomes are the values of the amounts of soda in ounces. It is important to appreciate that the sample is obtained from a population of bottles of soda that are sold by the business establishment. In this case, the mean of the sample would be obtained via dividing the sum of the values of the amounts of soda in the bottles by the sample size, i.e. 30.
The median of a sample is a value that is at the center when the values are sorted in an ascending manner (Heiman, 2013). With regard to the 30 values in this case, they would be arranged from the smallest to the largest. However, because the sample size (30) is an even number, it would be expected that two values would be in the middle. Thus, the values would be added and divided by two to obtain their average, which would be the median.
Standard deviation is a value that is used in estimating the extent to which all values are close to the sample mean (Heiman, 2013; Spatz, 2010). Samples that are typified by small standard deviations could imply that they are tightly grouped. In addition, they could be termed as containing precise data. The following are the results of the mean, median and standard deviation that are calculated using SPSS version 16.0:
|Number of values||30|
Figure 1. A table showing the results of the mean, median and standard deviation.
Constructing a 95% confidence interval for the ounces in the bottles
A confidence interval is a range of values that contains a parameter of interest of a population (Heiman, 2013). In this context, the parameter of interest would be the sample mean. Thus, if several samples from the population of bottles are analyzed, then there would be an interval that would always contain the mean. A 95% confidence interval would imply that there would be a certainty level of 95% that the mean would always be within a certain interval. Based on the SPSS analysis in table 1, it is evident that the standard deviation of the sample is 0.5503, which would be used in constructing the 95% confidence interval as follows:
75% percentile (upper limit of the CI) = sample mean + (degrees of freedom) x (standard deviation) = 14.87 + (0.21 + 0.5503) = 15.08
25% percentile (lower limit of the CI) = sample mean – (degrees of freedom) x (standard deviation) = 14.87 – (0.21 + 0.5503) = 14.66
Thus, the 95% confidence interval ranges from 14.66 ounces to 15.08 ounces.
A hypothesis test
Hypothesis testing would be appropriate in determining the truth about the claims that bottles contain less than sixteen (16) ounces of soda. A hypothesis is an assumption that can be supported or disapproved through statistical testing (Spatz, 2010). An assumption would be stated as follows: There is no significant difference between the mean of values less than 16 ounces and the mean of values that are equal to or more than 16 ounces. A t-test would be used to compare the mean values. The following table shows the results:
|Table Analyzed||Data 1|
|Unpaired t test|
|P value summary||***|
|Are means significantly different? (P < 0.05)||Yes|
|One- or two-tailed P value?||Two-tailed|
|t, df||t=3.796 df=28|
|How big is the difference?|
|Mean ± SEM of column A||14.79 ± 0.08755 N=28|
|Mean ± SEM of column B||16.05 ± 0.05000 N=2|
|Difference between means||-1.264 ± 0.3331|
|95% confidence interval||-1.946 to -0.5821|
Figure 2. A table showing results of unpaired t-test.
As shown in the table, there is a significant difference between the mean values. Thus, the null hypothesis is not supported by the analysis. It is critical to conclude that the claims of consumers that the bottles of the company contain product that is less than 16 ounces are true.
Three possible causes could have led to the discrepancies in the firm. First, ineffective methods could have been used in sealing the bottles of soda. Therefore, some soda could have evaporated. Second, personnel involved in the final measurements of the product could have set the wrong values in the packaging machine. Third, the packaging machine could be faulty, leading to wrong amounts of soda in the bottles.
It would be important for the management of the soda bottling company to adopt some strategies that would eliminate the problem in the future. It would be recommended that the packaging machine be inspected and calibrated on a regular basis. Workers who operate the machine should be retrained on a regular basis. Lastly, external agencies should be involved to ensure that the firm complies with quality standards related to the standard volume of soda in the bottles.
Heiman, G. (2013). Basic statistics for the behavioral sciences. Boston, MA: Cengage Learning.
Spatz, C. (2010). Basic statistics: Tales of distributions. Boston, MA Cengage Learning.