Limits for a Single Mean

We have shown that the difference of the mean of this quantity (A—B) is not significantly different from zero. This attainment of non-significance might be due to the quantity not differing from zero, but alternatively it might be due to the inadequacy of our data, i.e. our data may be insufficient to give us a reasonably accurate estimate of the quantity. We can estimate which of these two alternatives is operating as follows. 

We are going to determine the limits between which we can be reasonably confident that the true value of the mean, as determined from our sample, lies. 

The first requirement is to decide what we mean by reasonably confident. We might consider that if we are right 19 out of 20 times (equivalent to 95%) we would be satisfied. 

We would then look np in the table of t for 9 degrees of freedom its value for the 5% level (95% chance of being right is equivalent to a 5% chance of being wrong). Here t is 2.26. Then the limits dzL on either side of the sample mean, where a = the standard deviation of the population and n is the number of individuals in the sample, are 

Thus it is reasonably certain (within that chosen probability level) that the true value of the mean lies between —1.10 -t- 1.28 = 0.18 and —1.10 — 1.28 = — 2.38. These limits are known as 95% confidence limits. 

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