Question

Humerus bones from the same species of animal tend to have approximately the same length-to-width ratios. When fossils of humerus bones are discovered, archeologists can often determine the species of animal by examining the length-to-width ratios of the bones. It is known that species A has a mean of 8.5. Suppose forty-one fossils of humerus bones were unearthed at an archeological site in East Africa, where species A is believed to have inhabited. (Assume that the unearthed bones are all from the same unknown species). The length-to-width ratios of the bones were measured and are summarized as follows: x̅=9.25 S=1.16 We wish to test the hypothesis that μ, 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, i.e., we wish to test whether the unearthed bones are from species A. suppose we also want a very small chance of rejecting H0. If in fact μ is equal to 8.5. That is, it is important that we avoid making a Type I error. The hypothesis-testing procedure that we have developed gives us the advantage of being able to choose any significance level that we desire. Since the significance level, α, is also the probability of a Type I error, we will choose α to be very small. In general, researches who consider a Type I error to have practical consequences should perform the test at a very low α-value-say, α=X. 01. Other researches may be willing to tolerate an α-value as high as .10 if a Type I error is not deemed serious error to make in practice. Test whether μ, the population mean ratio, is different from 8.5, using a significance level of α= .01.

Answer 1

Given: = 8.5, s = 1.16, = 9.25, n = 41, = 0.01

The Hypothesis:

H0: = 8.5

Ha: 8.5

This is a two tailed Test.

The Test Statistic: We use the t test statistic, as the population standard deviation is unknown. The test statistic is given by the equation:

t observed = 4.14

The Degrees of freedom: df = n - 1 = 41 - 1 = 40

The Critical Value: The Critical value at = 0.01, df = 40 is +2.704 and -2.704

The p Value: The p value (Two Tail) for Z = 4.14, df = 40 is; p value = 0.0002

The Decision Rule: If t observed is < -tcritical or if t observed is > tcritical, then reject H0.

If P value is < , Then Reject H0.

The Decision: Since t observed (4.14) is > 2.704, we Reject H0.

Since P value (0.0002) is < (0.01) , We Reject H0.

The Conclusion: There is sufficient evidence at the 99% significance level to conclude that the population mean ratio of all bones of this particular species is different from 8.5.

________________________________________________________________________