In this case study, I as a manager of a bottling company should perform statistical analysis of the average amount of soda in the bottles produced by this company. There are thirty observations were pulled and the volume of soda was recorded for each bottle.
Step #1. Calculating the descriptive statistics (mean, median and standard deviation).
This method (descriptive statistics) is used for general review and description of the data. It usually is the initial step in the analysis of quantitative data and the use of other statistical procedures. The features of the sample data can serve as a basis for conclusions about the characteristics of the entire population with a given level of confidence. The method offers an effective and relatively simple way of reviewing and description of the data, as well as a convenient way of presenting such information. In particular, graphical methods are very convenient for the presentation and transmission of information. Descriptive statistics is applicable in all cases of the use of data and can be useful in the analysis of a decision-making process. In this part, we calculate three measures of central tendency and variability for the volume of soda in the bottles. The calculations are completed in Excel and the results are given in the table below:
After calculating the descriptive statistics we can conclude the following: the average volume of soda in the bottles produced by the company is 15.85 ounces with the standard deviation of 0.661 ounces. The “middle” element (that divides 50% bottom and 50% top parts of the data) is 15.99 ounces. The mean value is approximately equal to the median value, indicating that the distribution is approximately symmetric.
Step #2. Calculating the 95% confidence interval for the mean value.
The formula for the 95% CI has the following form:
x±1.96sn
Here, x is the sample mean, s is the sample standard deviation and n is the sample size. The value of 1.96 is the so-called 95% percentile (z-value for the level of confidence of 95%). It is taken from the tables of the standard normal distribution. I calculate the lower 95% limit and the upper 95% limit in Excel. According to the calculations, I am 95% confident that the average amount of soda in the bottles is between 15.62 ounces and 16.09 ounces.
Step #3. Hypothesis test.
There are two different tests that compare the mean values of the sample with the population mean value: z-tests and t-tests. Z-test is used when researchers know the population standard deviation and t-test is used when we do not know the standard deviation of the population. For t-test, the population standard deviation is estimated using the sample standard deviation and the sample size.
Since the population standard deviation for the average amount of soda in the bottles is unknown, I use t-distribution. One-sample t-test is used to examine the hypothesis that the average number of ounces of soda in the bottles is approximately 16 ounces. Thus, the null hypothesis is: the average volume of soda in the bottles is not significantly different from 16 ounces. The alternative hypothesis is: the average amount of soda in the bottles is significantly less than 16 ounces. The test is one-tailed. The level of significance is assumed at 0.05.
t=x-μsn=15.85-160.66130=-1.243
The degrees of freedom in this test are:
df=n-1=30-1=29
The critical value of the test is:
t0.05, 29=1.699
Since 1.243<1.699, I don’t reject the null hypothesis. At the 5% level of significance, I state that the mean value of the soda amount in the bottles (based on this sample) is not significantly different from the advertised 16 ounces. I conclude that the company fills the bottles accordingly. The alternative hypothesis is accepted.
Step #4. Discussion and conclusion.
The results of the statistical test do not give me enough evidence to doubt the fact that the company fills the bottles in accordance to their advertised amount of soda. I think that the company has received complaints from the customers, because these customers were from those part who bought the bottles with the least amount of soda. For example, there are some bottles with the amount of soda less than 15 ounces. Naturally, these clients have complained. However, those customers who bought the bottles with almost 17 ounces of soda, probably haven’t sent a claim to the company. The issue can be explained by a relatively high standard deviation of the average volume of soda in the bottles. I can recommend the strategy that is aimed at the reduction of the variance of soda in the bottles. For example, the company should calibrate the equipment to reduce the scattering of a dose of liquid dispensed in the bottles. This helps deal with the situation where there is one bottle of about 14 ounces of soda and the other is of almost 17 ounces.
