Sec. 6.3 ESTIMATION OF /* AND ff 2 167 



following results will be assumed to have been obtained: x — 1400 

 hours and s^ = 70 hours. The inequalities above become the follow- 

 ing after simplification: 1260 < n < 1540, 1242 < m < 1558, and 1172 

 < ix < 1628 hours, respectively, if the computations are rounded to 

 the nearest whole hour. If you act on the assumption that the true 

 average life of this type of bulb is between 1260 hours and 1540 hours, 

 you run a risk of 8 in 100 that the sample has misled you. However, 

 if the widest limits, 1172 to 1628, are used, the risk of an erroneous 

 assumption is only 1 in 100. 



Table 6.32 has been included to illustrate further and to clarify 

 the idea of confidence intervals. It contains some sampling results 

 obtained from a normal population with fx — 60, and the n = 10. A 

 summary of 578 samples is shown at the bottom of the table. Not 

 all the sampling results are given; just enough to satisfy the pur- 

 poses of this discussion. The asterisks indicate those intervals which 

 fail to include fi. 



Some of the points which are illustrated by Table 6.32 are the 

 following: 



(a) The confidence coefficients are long-run relative frequencies 

 which are verified only after a large number of samples. If atten- 

 tion were confined to samples 4 to 10, the confidence coefficients 

 would seem to be wrong; but over the set of 578 confidence intervals, 

 they are verified quite satisfactorily. 



(6) The determination of a confidence interval is doubly depend- 

 ent on chance: once as regards the mean, and again regarding the 

 magnitude of the standard deviation. For example, samples 306 and 



307 had essentially the same mean but the standard deviations were, 

 by chance, so different that even the 80 per cent limits from sample 

 306 included the true mean, 60. Only the 95 per cent confidence 

 interval from sample 307 includes ju. On the other hand, samples 

 303 and 308 have practically the same standard deviation, but the 

 sample means are so different that the 80 per cent limits from sample 



308 failed to include the true mean. 



(c) The confidence interval is wider for the larger confidence co- 

 efficients, that is, the more certain we choose to be in our conclu- 

 sions, the more room we must leave for sampling variations. 



Problem 6.31. Suppose that a highway commission is interested in the 

 strength of concrete which it wishes to make for highway projects, and that it 

 concludes that the 7-day tensile strengths of standard samples will be the best 



