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(a) For the numbers 116.0, 97.9, 114.2, 106.8, and 108.3, use the critical values for the Q-test (Table 7.5 in your textbook) to decide whether the number 97.9 should be discarded with 95% confidence.

(b) A new analytical method is developed, and the following analytical results were obtained: 2.11, 2.09, 2.05, 2.07, 1.96, 2.10, 1.94, 2.06, 1.93 ppm (N = 9).

The older method run on the same standard reference material provided the following results: 2.02, 1.94, 2.02, 1.98, 1.96, 1.98, 2.02, 2.01, 1.93, 1.99 ppm (N = 10).

Run a statistical test on the data sets to determine whether the new method has better precision.

Answer :

(a) For the numbers 116.0, 97.9, 114.2, 106.8, and 108.3, the number 97.9 should be discarded using the Q-test at a 95% confidence level.

(b) The statistical test on the data sets indicates that the new method has better precision compared to the older method.

(a) Q-test for Outlier Detection:

The Q-test is used to detect outliers in a data set. At a 95% confidence level, the critical value for a data set of size N = 5 is 0.830. The Q-value is calculated as the absolute difference between the suspect value (97.9 in this case) and the value closest to it, divided by the range of the data set.

Q = |97.9 - 106.8| / (116.0 - 97.9) = 8.9 / 18.1 ≈ 0.491

Since the calculated Q (0.491) is less than the critical Q-value (0.830), the number 97.9 is not an outlier, and it should not be discarded from the data set.

(b) Comparison of Precision:

To compare the precision of the two methods, we can calculate the standard deviation (SD) for each data set. For the new method, the SD is approximately 0.063 ppm, and for the older method, the SD is approximately 0.038 ppm.

The smaller the standard deviation, the better the precision. As the SD for the new method (0.063 ppm) is larger than that of the older method (0.038 ppm), we can conclude that the older method has better precision in measuring the given standard reference material.

It's important to note that "better precision" does not necessarily imply "better accuracy." Precision refers to the consistency and reproducibility of results, while accuracy reflects how close the measured values are to the true or expected value.

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