Changing the multiplying constant

The formula for the 95 per cent confidence interval (and also for the 99 per cent confidence interval) given above is in fact not quite correct. It is correct up to a 

The reason for this is again a technical one but relates to the uncertainty associated with the use of the sample standard deviation (s) in place of the true population value (a) in the formula for the standard error. When a is knovm, the multiplying constants given earlier apply. When ct is not known (the usual case) we make the confidence intervals slightly wider in order to account for this uncertainty. When n is large of course s will be close to a and so the earlier multiplying constants apply approximately. 

A more complete table can be found in many standard statistics textbooks. Alternatively most statistics packages will contain a function that will give the multiplying constants for any value of degrees of freedom. 

In an asthma trial comparing two short acting treatments, the following (hypothetical) data were obtained for the increase in FEVj (Table 3.2)  

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There is a connection with what we are seeing here and the calculation of the confidence interval in Chapter 3. Recall Table 3.1 within Section 3.1.3, Changing the multiplying constant . It turns out that p-values and confidence intervals are linked and we will explore this further in a later chapter. The confidence coefficients for d.f. = 38 are 2.02 for 95 per cent confidence and 2.71 for 99 per cent confidence. If we were to look at the tjg distribution we would see that 2.02 cuts off the outer 5 per cent probability while 2.71 cuts off the outer 1 per cent of probability. 

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