ve then standard Equation deviations 1.45 of modified sample give the sizes standard of can be to error of the difference between means as in Equation 1.49. A sample consists of a relatively small number of items drawn from a much larger population. This sample is analysed for certain at- (1.49) tributes and it is then assumed that these attributes apply to the total population, within a certain tolerance of error. Sampling is usually associated with the normal probability dis- tribution and, based on this distribution, the errors which arise due to sampling can be estimated. Suppose a sample of n items is taken Similarly Equation 1.47 can be modified to give the standard error from a population of n items which are distributed normally. If the of the difference between probabilities of two samples from the sample is found to have a mean of Ms with a standard deviation of same population as in Equation 1.50. 1/10 Statistical analysis (1.50) independently. If the data are presented in the form of a table having V vertical columns and H horizontal rows then the degrees of freedom are usually found as in Equation 1.54. Degrees of freedom = (V - 1) (1.54) Tests of significance Returning to the earlier example, suppose a coin is tossed 100 times and it comes up heads 60 times and tails 40 times. Is the coin In taking samples one often obtains results which deviate from the biased? The expected values for heads and tails are 50 each SO that expected. Tests of significance are then used to determine if this expression 1.55 is obtained. deviation is real or if it could have arisen due to sampling error. (1.55) 1.11.1 Hypothesis testing In this system a hypothesis is set up and is then tested at a given The number of degrees of freedom is one since once we have fixed confidence level. For example, suppose a coin is tossed 100 times the frequency for heads that for tails is defined. Therefore entering and it comes up heads 60 times. Is the coin biased or is it likely that Table 1.3 with one degree of freedom the probabilit