In MATHSTATSOU/Intro2R: Introduction to R

knitr::opts_chunk$set(
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library(Intro2R)

Introduction

The first case study relates to the BONES data set. The information and questions come directly from our text, MS page 386-387

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 ratio of 8.5. Suppose 41 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 listed in Table 8.2. We wish to test the hypothesis that m, 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.

The questions concerning the data set are :

a. Suppose we want a very small chance of rejecting $H_0$, if, in fact, $\mu$ is equal to $8.5$. That is, it is important that we avoid making a Type I error. Select an appropriate value of the significance level, a. b. Test whether $\mu$, the population mean length-to-width ratio, is different from $8.5$, using the significance level selected in part a.

Solution

The text book uses a Z approximation assuming that $n=41$ validates the approximation (large sample).

You should realize by now that given the assumptions of the t test are upheld we do not need to be concerned about large/small sample issues. The t-test will be valid. The only assumption that might cause a problem when n is small is the assumption of Normality $\bar{Y}_i\stackrel{iid}{\sim}N$. If n is large $\bar{Y}_i\stackrel{iid}{\sim}N$ even if $Y_i$ is not normal, due to the CLT.

a,b

We will use the same $\alpha=0.01$ because we wish to have a small type 1 error.

The hypotheses are:

$$H_0:\mu=8.5\H_1:\mu\ne 8.5$$

R code

We will now carry out a one sample t test. Notice that by default this is a two sided test -- this is needed because of the alternate hypothesis $H_1:\mu \ne 8.5$

with(bones, t.test(LWRATIO, mu=8.5, conf.level=0.99))

Conclusion

We will reject the Null at $\alpha=0.01$ in favour of the alternate.

We now have statistical evidence that the bones do not come from species A.

MATHSTATSOU/Intro2R documentation built on Feb. 20, 2025, 6:18 a.m.