Humerus bones from the same species of animal tend to have approximately the same length-to-width ratios. It is known that species A has a mean of 8.5. Suppose 43 fossils of humerus bones were unearthed at an archaeological site in East Africa, where species A is believed to have inhabited. The length-to-width ratios of the bones were measured from the samples collected and are summarized: average 9.25, standard deviation 1.16. To test the hypothesis that the population mean ratio of all bones of this particular species, is equal to 8.5 against the alternative that it is different from 8.5. To avoid Type I error, we choose the significance level alpha at 0.01. What is the z value of this particular test?"

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...