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Since the sample mean and variance are independent, and the sum of normally distributed variables is also normal, we get that: ^ + ˙ (+, + ()) Based on the above, standard confidence intervals for + can be constructed (using a Pivotal quantity) as: ^ + + And since confidence intervals are preserved for monotonic transformations, we get that
Without the use of either trigonometry or a table of chords, Archimedes determines the Sun's apparent diameter by first describing the procedure and instrument used to make observations (a straight rod with pegs or grooves), [58] [59] applying correction factors to these measurements, and finally giving the result in the form of upper and lower ...