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Descriptive Analysis

A quality characteristic of interest for a tea-bag-filling process is the weight of the tea in the individual bags. If the bags are underfilled, two problems arise. First, customers may not be able to brew the tea to be as strong as they wish. 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 5.5 grams of tea in a bag. If the average amount of tea in a bag exceeds the label weight, the company is giving away product. Getting an exact amount of tea in a bag is problematic because of variation in the temperature and humidity inside the factory, differences in the density of the tea, and the extremely fast filling operation of the machine (approximately 170 bags a minute). The following table provides the weight in grams of a sample of 50 tea bags produced in one hour by a single machine.

[See attached Table to answer the questions below].

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
tea bags be concerned about the central tendency?
e. Interpret the measures of variation within the context of this problem. Why should the company producing the tea bags
be concerned about variation?
f. Construct a box-and-whisker plot.
g. Are the data skewed? If so, how.
h. Is the company meeting the requirement set forth on the label that, on average, there are 5.5 grams of tea in a bag?
i. If you were in charge of this process, what changes, if any, would you try to make concerning the distribution of weights in the individual bags?

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A quality characteristic of interest for a tea-bag-filling process is the weight of the tea in the individual bags. If the bags are underfilled, two problems arise. First, customers may not be able to brew the tea to be as strong as they wish. 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 5.5 grams of tea in a bag. If the average amount of tea in a bag exceeds the label weight, the company is giving away product. Getting an exact amount of tea in a bag is problematic because of variation in the temperature and humidity inside the factory, differences in the density of the tea, and the extremely fast filling operation of the machine (approximately 170 bags a minute). The following table provides the weight in grams of a sample of 50 tea bags produced in one hour by a single machine.

5.65 5.44 5.42 5.40 5.53 5.34 5.54 5.45 5.52 5.41
5.57 5.40 5.53 5.54 5.55 5.62 5.56 5.46 5.44 5.51
5.47 5.40 5.47 5.61 5.53 5.32 5.67 5.29 5.49 5.55
5.77 5.57 5.42 5.58 5.58 5.50 5.32 5.50 5.53 5.58
5.61 5.45 5.44 5.25 5.56 5.63 5.50 5.57 5.67 5.36

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a. Compute the arithmetic mean and median.

Mean: The mean is the average of all the numbers. You calculate it by adding all the numbers together and dividing by the number of observations:

mean = 275.07/50 = 5.5014

Median: The median is the middle number. You find it by sorting all the numbers from smallest to largest, then taking the number in the middle of the list. When there is an even number of observations (like here), there is no "middle" number. In that case, you take the average of the two middle numbers. You can do this by hand if you want, but 50 numbers is a long ...

Solution Summary

The solution includes answers to the question with detailed explanations, including how to calculate the mean, median, standard deviation, range, variance, coefficient of variation, and other statistics.

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