168 SAMPLING NORMAL POPULATIONS Ch. 6 



criterion of quality. Suppose also that ten of the standard testing models gave 

 these results: 



x = 439.0 pounds per square inch, s = 47.0 pounds per square inch. 



What valid and useful conclusions could they draw concerning the true average 

 tensile strength of this concrete? 



Although the true average strength, /x, is a hypothetical strength 

 rarely possessed by an actual sample, it does provide a useful de- 

 scription of the tensile strength of a type of concrete. Before a con- 

 fidence interval can be put on p. a confidence coefficient must be 

 chosen. Such matters as the seriousness of committing an error, and 

 the added cost of demanding narrower limits, are involved in this 

 decision. However, for purposes of illustration it will be assumed 

 that a risk of 1 in 20 of obtaining a confidence interval not including 

 [x is appropriate to these circumstances. Then, using inequality 

 (6.37) because n = 10 and 95 per cent limits are sought, we obtain 

 the following: 



439 - 2.26(14.9) < jlc < 439 + 2.26(14.9) 



because s £ = 47.0/v 10 = 14.9 pounds per square inch. When this 

 inequality is simplified it is found that the 95 per cent confidence inter- 

 val is 



405 pounds per square inch < n < 475 pounds per square inch, 



to the nearest 5 pounds. Therefore, the true average tensile strength 

 of this concrete will be considered to be somewhere between 405 and 

 475 pounds per square inch; but, at the same time, it will be kept 

 in mind that there is 1 chance in 20 that this sample has been "wild" 

 and hence has led to an incorrect conclusion. 



If the reader thinks a bit about the material in this section as 

 compared to the corresponding section in the preceding chapter on 

 binomial populations, it should become apparent that these two sec- 

 tions have a great deal in common. In both, a sampling distribution 

 was studied, and we were concerned with the relative frequencies 

 with which certain sampling phenomena would occur. In particu- 

 lar, we were interested in the relative frequencies with which inter- 

 vals determined from samples would include the unknown population 

 parameter. This probability is the confidence coefficient. 



There also are some differences which could be pointed out. A 

 major one is that owing to the discontinuity of the binomial frequency 

 distribution, the confidence coefficient is the lower limit on the rela- 



