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# Arithmetic mean, median and probablity

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1. A property of concern for any food company that uses a high-speed carton-filling machine to package juice is the weight of the food product in the individual cartons. If the cartons are under filled, two problems arise. First, customers may not have enough product for their needs. Second, the company may be in violation of the truth-in-labeling laws. In this example, the label weight on the package indicates that, on average, there are 2.5 ounces of product in a carton. If the average amount of product in a carton exceeds the label weight, the company is giving the product for free. Getting an exact amount of product in a carton is problematic because of variation in the temperature and humidity inside the factory, differences in the density of the product, and the extremely fast filling operation of the machine (approximately 450 cartons per minute). The following table provides the weight in ounces of a sample of 60 cartons produced in one hour by a single machine:

4.01 4.06 2.45 2.06 2.02 2.59 2.72 3.08 3.08 3.04
2.22 2.47 2.96 2.41 2.42 2.09 3.03 3.09 1.98 3.05
3.11 2.31 1.28 3.01 2.42 2.49 1.57 2.46 2.55 2.52
3.02 1.87 1.99 1.88 1.38 3.06 3.04 3.04 3.07 2.52
3.08 3.03 1.62 2.32 1.43 2.12 2.43 2.99 1.85 3.08
3.04 3.11 1.59 1.81 3.02 2.99 3.01 1.76 3.01 2.33

a. Compute the arithmetic mean and median.
b. Compute the first quartile and third quartile.
c. Compute the range, interquartile range, variance, standard deviation, and coefficient of variation.
d. Interpret the measures of central tendency within the context of this problem. Why should the company producing the bottles be concerned about the central tendency?
e. Interpret the measures of variation within the context of this problem. Why should the company producing the bottles be concerned about variation?