180 SAMPLING NORMAL POPULATIONS Ch. 6 



sidered separately. The reader should verify the fact that if in for- 

 mulas 6.53 and 6.54 n x = n 2 = n, these formulas become 6.51 and 

 6.52, respectively. 



Many other applications of the ^-distribution, and accompany- 

 ing statistical techniques, could be cited; but the fundamental prin- 

 ciples are essentially the same as those already explained. 



PROBLEMS 



1. Suppose that 5 experimental concrete cylinders of each of two types of 

 concrete have been tested for breaking strength, with the following results in 

 hundreds of pounds per square inch: 



Type 1 : 40, 50, 48, 46, and 41 ; and 

 Type 2: 65, 57, 60, 70, and 55. 



Use the ^-distribution to determine if the difference in average breaking strength 

 between the two types of concrete can be assigned reasonably to mere sampling 

 variation. 



2. Suppose that two groups of 10 steers have been fed two different rations 

 (one to each group) and that the steers are of the same age, breed, and initial 

 weight. Given the following computations determine the 99 per cent confidence 

 interval on the true difference between the means of the average daily gains 

 under the two rations: 



Ration A Ration B 



m = 10 w 2 = 10 



xi = 1.90 lb/day x 2 - 1-55 lb/day 



s = 0.20, 18 D/F; t = 3.92 



Ans. CI 99 : 0.1 < I mi - m I < 0.6 lb/day. 



3. Suppose that an experiment has been set up at an engineering laboratory 

 to determine the difference in average breaking load between oak and fir beams 

 of the dimensions: 2 inches x 2 inches x 28 feet. The data from tests on 10 

 beams of each wood are as follows, in pounds: 



Oak: 725, 1015, 1750, 1210, 1435, 1175, 1320, 1385, 

 Fir: 1205, 810, 1110, 530, 765, 1075, 1475, 950, 



Oak: 1505, and 1340. Sum = 12,860: 2X 2 = 17,243,550. 

 Fir: 1020, and 1070. Sum = 10,010: 2X 2 = 10,625,400. 



If you can afford a risk of an error of only 1 in 100 what confidence limits do you 

 set on the true difference in average breaking load for these two materials? 



4. Draw 5 pairs of samples, each with n = 10, from the laboratory population 

 furnished you by the instructor, and compute t = d/sa for each pair of samples. 

 Then obtain from Table IV the probability that a numerical value of t that size 

 or larger would be obtained while pairs of samples are drawn from the same normal 

 population. 



