Problem 4: Equipment Precision Comparison Suppose you are trying to make a difficult measurement. Fortunately there is commercial equipment available for this purpose, although it is expensive. Your company has a large budget and wants to obtain the best equipment, but it also does not want to waste money needlessly. You are responsible for performing some tests to guide their decision. You have ordered two trial samples of metering equipment to test which one is better: Equipment A (which costs £60,000) and Equipment B (which costs £30,000). You take 10 measurements using each in a controlled environment. Equipment A gives the following readings: 128.00, 125.04, 125.17, 128.62, 126.06, 124.54, 128.80, 129.98, 126.49, 127.16 Equipment B gives: 122.16, 127.35, 124.73, 129.51, 123.60, 132.67, 131.07, 126.20, 132.44, 126.91 You may assume that measurement errors are normally distributed. 1. The "correct" value for the measurement is supposed to be 127. Verify that both tools are properly "calibrated" (i.e., that they provide measurements that on average are consistent with this value) with an appropriate statistical test. 2. Suppose you did not know that the true value was 127, or there was a possibility that the true value was not 127. Use a statistical test to evaluate whether the two tools produce measurements that are, on average, consistent with each other. 3. Company specifications require that the calibration accuracy (the absolute difference between the average of a very large number of measurements and the "correct" value) of the tool must be better (less) than 5. Show that both tools meet this requirement to better than 99% confidence under the assumptions above. 4. The most important consideration in your decision is precision: you want the tool that produces measurements with the least variance (lowest standard deviation). Can you tell (using an appropriate statistical test) if one tool is significantly more precise than the other? If so, which tool? Quote a p-value, and use a confidence interval to quantify how much more precise one tool is (or isn't) than the other. 5. Tool A is much more expensive, and your company might not want to spend the extra money if it cannot shown to be clearly superior. Conduct a modified version of the above hypothesis test with this information in mind, and quote a new p-value. 6. Would you recommend purchasing tool A, tool B, or would you run more tests (at a cost of £5,000 in overheads plus £500 per test)? If you run more tests, how many more tests would you run? Explain the basis for your decision in a few sentences or less.