Have you ever wondered, when you weigh yourself on a scale, how the scale is calibrated? How is it known that 150 pounds on that scale is the same as 150 pounds on another? Scales are calibrated by comparison to standard weights. Of course, millions of instruments require calibration and they cannot all be compared to the same weight. Rather, there is a single definitive weight and a hierarchy of other weights that are compared to it. The US National Bureau of Standards maintains a set of standard weights that are an important link in this chain.

In Statistics, Freedman et al. report the results of a set of definitive measurements by the Bureau of one 10 g standard weight, done sometime in 1962–1963. The first measurement was 9.999591. The error is very slight, on the order of what a grain of salt weights. Several other measurements also came in just slightly below 10 g. To make interpretation easier, the Bureau chose to measure not “grams,” but rather “micrograms below 10 g.” A microgram is one-millionth of a gram. So, instead of 9.999591, the measurement was 409. Here is a sample of 20 such measurements, all of this same weight: 409, 400, 406, 399, 402, 406, 401, 403, 401, 403, 398, 403, 407, 402, 401, 399, 400, 401, 405, 402

(a) Based on this sample, you would estimate that this standard weight falls short by _____ micrograms.

(b) The variation between one measurement and another is probably due to __________.

(c) Do a bootstrap simulation to determine how much variability there might be in your estimate in part (a). Produce a confidence interval. (This should be done on computer. If you have so far been totally unsuccessful in getting any of the software programs to work, you may do 10 simulations with actual slips of paper and a hat, report the results, and describe how you would use the results of 1000 such simulations).

(d) (OPTIONAL) Calculate this interval using statistical software and a standard formula approach.