New PDF release: An Introduction to Econometrics

By A. A. Walters (auth.)

ISBN-10: 0333025679

ISBN-13: 9780333025673

ISBN-10: 1349152773

ISBN-13: 9781349152773

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FIG. 3·3· Sampling Dbtn"bution of Means with Two Alternative Hypotheses Intuitively this comparison of the odds gives us a way of choosing between competing hypotheses. If the sample is taken, we simply calculate the sample average and then inspect the odds. We then choose the hypothesis which is favoured by the odds and reject the alternative hypothesis. Thus if the sample average is C we reject I in favour of II. If, however, the sample average is F we reject II in favour of I. If we obtain a sample average of J, then we merely cannot choose t Strictly one should talk about the probability of a sample value falling in an infinitesimal range about the value C, then the probability is measured by the rectangle of infinitesimal width and of height BC.

It is, however, not efficient for many populations. 38 METHODOLOGY, PROBABILITY AND SAMPLING Small Samples One main comfort of having a large sample is derived from the fact that we may assume that the estimate of the standard deviation from the sample is very close to the true standard deviation. With large samples we do not need to worry about the fact that the standard deviation of the sample is only an estimate of the true standard deviation; the estimate is always near enough to the true value to make our tests and estimates useful.

Thus if we say 'the hypothesis is rejected' when t exceeds t. 011 or is smaller than - t. 05 we shall be rejecting a true hypothesis in only 1 out of 20 cases. These are exactly analogous to the statements made with large-sample theory. Since the t distribution varies according to the number of degrees of freedom it is obviously convenient to have key values tabulated. Included in any set of standard tables one may find t. 011 tabulated for various numbers of degrees of freedom. As the degrees of freedom increase, the value of t.

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An Introduction to Econometrics by A. A. Walters (auth.)

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