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Measures of Central Tendency and Variation

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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-labelling laws. For this product, the label weight on the package indicates that, on average, there are 5.5 grams of tea in a bag. If the mean 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 per minute). The data shown below are weights, in grams, of a sample of 50 tea bags produced in one hour by a single machine:

5.45 5.61 5.44 5.25 5.56 5.63 5.50 5.57 5.67 5.36
5.57 5.40 5.53 5.54 5.55 5.62 5.56 5.46 5.44 5.51
5.65 5.44 5.42 5.40 5.53 5.34 5.54 5.45 5.52 5.41
5.47 5.40 5.47 5.61 5.53 5.32 5.67 5.29 5.49 5.55
5.70 5.57 5.42 5.58 5.58 5.50 5.32 5.50 5.53 5.58

a. By using Microsoft Excel for computation, list down the values of:
(i) mean, median, minimum, maximum, first quartile, and third quartile.
(ii) range, variance, and standard deviation.

b. Interpret the measures of central tendency and variation within the context of this problem.
Why should the company producing the tea bags be concerned about the central tendency and variation?

c. By using the results in (a), construct a boxplot. Indicate clearly on the boxplot the five-number summary.
Are the data skewed? If so, describe its skewness.

d. Is the company meeting the requirement set forth on the label that, on the average, there are 5.5 grams of tea in a bag?
If you were in charge of this process, what changes, if any, would you try to make concerning the distribution of the weights in the individual bags?

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