1. Two thermocouples (temperature measurement devices) are tested by inserting their probes in boiling water, recording the readings, removing, and drying the probes, and then doing it again. The results of five measurements are as follows:

T(°C)—Thermocouple A 72.4 73.1 72.6 72.8 73.0
T(°C)—Thermocouple B 97.3 101.4 98.7 103.1 100.4
For each set of temperature readings, calculate (using Excel) the sample AVERAGE, the RANGE, and the sample STANDARD DEVIATION.

2. Your company manufactures plastic wrap for food storage. The tear resistance of the wrap, denoted by X, must be controlled so that the wrap can be torn off the roll without too much effort, but it does not tear too easily when in use. In a series of test runs, 15 rolls of wrap are made under carefully controlled conditions and the tear resistance of each roll is measured. The results are used as the basis of a quality assurance specification. If X for a subsequently produced roll falls more than two STANDARD DEVIATIONS away from the test period average (AVERAGE), the process is declared out of specification and production is suspended for routine maintenance. The test series data are as follows:

Roll 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
X 134 131 129 133 135 131 134 130 131 136 129 130 133 130 133

(a) Write an Excel spreadsheet to take as input the test series data and calculate the sample AVERAGE and sample STANDARD DEVIATION, using built-in functions for the calculations.

(b) The following tear resistance values are obtained for rolls produced in 14 consecutive production runs subsequent to the test series: 128, 131, 133, 130, 133, 129, 133, 135, 137, 133, 137, 136, 137, 139. On the Excel spreadsheet and using the Excel plotting capability, plot a control chart of X versus run number, showing horizontal lines for the values corresponding to (AVERAGE), (AVERAGE-2×STANDARD DEVIATION), and (AVERAGE+2×STANDARD DEVIATION) from the test period, and show the points corresponding to the 14 production runs. Which measurements led to suspension of production?

(c) Following the last of the production runs, the chief plant engineer returns from vacation, examines the plant logs, and says that routine maintenance was clearly not sufficient and a process shutdown and full system overhaul should have been ordered at one point during the two weeks he was away. When would it have been reasonable to take this step, and why?

3. Product quality assurance (QA) is a particularly tricky business in the dye manufacturing industry. A slight variation in reaction conditions can lead to a measurable change in the color of the product, and since customers usually require extremely high color reproducibility from one shipment to another, even a small color change can lead to rejection of a product batch. Suppose the various color frequency and intensity values that comprise a color analysis are combined into a single numerical value, C, for a particular yellow dye. During a test period in which the reactor conditions are carefully controlled and the reactor is thoroughly cleaned between successive batches (not the usual procedure), product analyses of 12 batches run on successive days yield the following color readings: