Have you ever wondered how atomic weights are determined? The National Institute of Standards and Technology (NIST) is one of several research organizations that seeks to make accurate measurements of this sort. The most common approach is to prepare a high-purity sample of the element by extracting it and purifying it from a natural source. The mass of the element’s different isotopes and their isotopic abundance is then determined using a high resolution mass spectrometer. The problem with this approach is that the purification process may affect the relative isotopic abundances, providing a source of determinate error. The data in this problem come from a 1985 NIST study of gallium in which they measured the relative abundance of two isotopes: ^{69}Ga and ^{67}Ga.

(a) NIST has a reference standard that they analyzed a sufficient number of times over the preceding 20 years such that its value of 1.52831 for the ^{69}Ga/^{67}Ga ratio is well known and can be treated as μ. In the 1985 study they reanalyzed this sample, reporting the ratio as 1.52841 with a standard deviation of ±0.00011 for seven determinations. Is there any evidence at α = 0.01 to suggest that the 1985 results have a determinate error?

(b) As part of the 1985 study, NIST conducted single analyses on 16 different samples of purified gallium (each of these samples came from a different source, and one of the samples was the NIST reference sample described above). The ratio of ^{69}Ga to ^{67}Ga for these samples are tabulated below.

1.52729 1.53017 1.52646 1.52775 1.52954 1.52857 1.52729 1.52841

1.52696 1.52580 1.52559 1.52549 1.52569 1.52620 1.52612 1.52796

Report the mean, median, standard deviation, variance, and 95% confidence interval for this data set.

(c) What is the probability that a randomly collected sample of gallium will have a ^{69}Ga/^{67}Ga ratio between 1.52808 and 1.52874 (i.e. the value determined for the NBS standard ± three standard deviations)?

(d) Now, consider your answers to part (a), which gives the variance for the method, and for part (b), which gives the combined variance due to the method and the collection of samples. Is there any evidence at α = 0.05 to suggest that the variance introduced by sampling is more significant than the variance introduced by the analytical method? What implication does your answer have to the determination of atomic weights? Based on this data, to how many significant figures do you think the atomic weight of Ga can be determined? Be sure to justify each of your answers.