y level for x2= this falls within a reasonable sampling error? The hypothesis is set 4 is seen to be above 2.5% i.e. there is a strong probability that the up that the coin is not biased. Therefore one would expect that the difference in the two results arose by chance and the coin is not probability of heads is 0.5, i.e. Ps = 0.5. The probability of tails, qs biased. is also 0.5. Using Equation 1.47 the standard error of probability is As a further example suppose that over a 24 hour period the given by Equation 1.51. average number of accidents which occur in a factory is seen to be (1.51) Table 1.3 The Chi-square distribution Therefore from Equation 1.48 the population probability at the 95.45% confidence level of getting heads is 0.5 + 2 x 0.05 = 0.6. Degrees Probability level Therefore it is highly likely that the coin is not biased and the results 0.100 0.050 0.025 0.010 0.005 are due to sampling error. freedom The results of any significance test are not conclusive. For example, is 95.45% too high a confidence level to require? The higher the confidence level the greater the risk of rejecting a true 10.60 hypothesis, and the lower the level the greater the risk of accepting 11.34 12.84 a false hypothesis. Suppose now that a sample of 100 items of production shows that 11.14 13.28 14.86 five are defective. A second sample of 100 items is taken from the 11.07 12.83 15.09 16.75 same production a few months later and gives two defectives. Does 10.64 12.59 14.45 16.81 18.55 this show that the production quality is improving? Using Equation 1.50 the standard error of the difference between probabilities is 12.02 14.07 16.01 18.48 20.28 given by expression 1.52. 13.36 15.51 17.53 20.09 21.96 (1.52) 14.68 16.92 19.02 21.67 23.59 15.99 18.31 20.48 23.21 25.19 18.55 21.03 23.34 26.22 28.30 This is less than twice the difference between the two prob- 21.06 23.68 26.12 29.14 31.32 abilities, i.e. 0.05 - 0.02 = 0.03, therefore the difference is very likely to have arisen due to sampling error and it does not necessar- 