Abstract Inference [math. stats.] by Ulf Grenander

By Ulf Grenander

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The question therefore arises as to what form the distribution of a difference between two sample means takes. We shall assume that both populations from which the two samples are taken have normal distributions and write the mean and standard deviation )11 and a 1 for the first population, and )12 and a 2 for the second population. If the sample sizes are n 1 and n2 then the means and variance of the distribution of the sample means are )11 and )12' and ai/n 1 and a~/n2' respectively. Here we must again quote a mathematical result without proof, that the quantity (Xl - x2 ) is normally distributed with mean ()11 - )12) and variance (ai/n 1 + a~/n2)' It is only reasonable to expect that the mean of the distribution of (Xl - x2 ) should be ()11 - )12)' At the first sight, however, the addition of the variance is perhaps surprising.

B) Three farmers use different methods of fattening pigs to bacon weight. Using similar pigs they record the number of days from weaning to bacon weight for each pig as follows Farmer C Farmer D Farmer E 105 112 99 97 104 107 107 108 104 112 101 103 105 108 100 107 100 113 103 115 98 110 105 Comparison of samples 35 Calculate the mean and variance for each sample and use a t-test to compare each farmer's results with the average local figure of 110 days from weaning to bacon weight. Obtain 95% confidence intervals for the mean number of days for each farmer and comment on the results.

The F-test is an overall test of the significance of the differences that have been observed between the means of all the treatments in the experiment. If the Testing overall variation between the treatments 51 o 2 3 4 5 Fig. s. when treatment effects are present. F-value is not significant then we must proceed with some caution in looking at individual differences between means. Tests of differences between pairs of means where the differences to be tested are chosen solely because they are large are clearly invalid.

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