By Maurice G. Kendall, William R. Buckland
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Extra info for A dictionary of statistical terms
When the null hypothesis is true, the standard deviation of the differences between any individual and the true mean = is given by The standard error for the difference in means should, therefore, just be divided by the square root of N. If we normalize the difference observed by this standard error-like normalizing factor we derive the critical ratio. The critical ratio is also a random variable, however, and it must, therefore, be represented by a distribution. This new distribution is the tool we use to associate the noise units to the probability measure we are actually seeking.
E. 001. Based on these data, the chance of seeing a sample mean 'this much' greater than 200, given the null hypothesis is true, is much less than 5%, and we must conclude that while the 'holiday therapy' decreased serum cholesterol levels significantly when compared with those of the general population, it did not lower the levels far enough, at least after three months, to enable the treated patients to be declared 'healthy'.
In either case you can easily reject the null hypothesis for these data. You can thus conclude, with at most a 5% chance of being wrong, that you would not be able to observe a mean serum cholesterol of 209 mg/dl if your patients were drawn from a population with a true mean and standard deviation of 237 mg/dl and 20 mg/dl respectively. The zstatistic and the one-sided critical value are shown schematically in Figure 17. Inference 51 Figure 17 Gaussian distribution The 'bell-shaped' curve depicts the standard normal or Gaussian distribution.