In a study of mating calls in the tree toad Hyla ewingi, Littlejohn (1965) found

Question

In a study of mating calls in the tree toad Hyla ewingi, Littlejohn (1965) found the note duration of the call in a representative sample of 39 observations (one from each of 39 toads) from Tasmania to have a mean of 189 ms and a standard deviation of 32 ms; the sample histogram followed the normal curve pretty well (a) Set up a statistical model for this situation, being explicit about the population, sample and imaginary data sets, and use your model (including the usual inferential summary) to build a 95% confidence interval for the population mean mu (what is the population here, precisely?). (b) Other scientists working in south-eastern Australia had previously found that the mean mating call duration for this species was 210 ms. (i) Does the difference between the Tasmania and Australia findings seem large to you in practical terms? Explain briefly. (ii) Is this difference large in statistical terms? Answer this question both by reference to your confidence interval in (a) and by calculating the P value for testing the hypothesis that the mean in Tasmania is 210 ms against the alternative that it's not. (c) When gathering data of this type, which of the following designs do you think would give the most information about mating calls for Tasmanian tree toads, or would they be equally informative, or would one design be better for some kinds of information and another design for other kinds (specify briefly in each case)? Explain briefly. (i) One observation from each of 39 toads; (ii) 39 observations from a single toad; (iii) 5 observations from each of 7 toads and 4 from an eighth toad.

Explanation / Answer

(b) (i) Here population mean in australia = 210 ms

In practical terms, we can calculate it by evaluating effect size.

Sample mean xbar = 189 ms ; Standard deviation = 32 ms

Effect Size = (210 - 189)/32 = 0.66

so effect size is large enough so we can say that in practical terms, difference between Tasmania and AUstraila are large enough.

(b) Now , we will check it statistically byusing t- test procedure and hypothesis testing.

Here,

Null Hypothesis : H0 : Mean in tasmania is equal to 210 ms. = 210 ms

ALternative Hypothesis : Ha : Mean in tasmania is not equal to 210ms. 210 ms

so standard error of sampling se0 = s/ n = 32/ 39 = 5.124

Test Statistic:

t = (xbar - )/se0 = (189 - 210)/ 5.124 = -4.1

so P - value = 2 * Pr (xbar <=189 ; 210; 5.124) = 0.0002

and confidence interval for population mean = xbar +- t0.025,38 (s/ n)

= 189 +- 2.0244 * 5.124 = (178.63, 199.37)

so we can say by both method that we can reject the null hypothesis and can conclude that population mean in tasmania is not equal to 210 ms.

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