156 SAMPLING NORMAL POPULATIONS Ch. 6 



The frequency distribution in the right-hand part of Table 6.21 is an 

 approximation to an infinite population of X{ which would result if this 

 sampling with n = 10 observations were continued indefinitely. Any 

 one random sample from the normal population described above neces- 

 sarily would be a member of the population of Xi. 



As an illustration of the preceding discussion, suppose that an agri- 

 cultural economist is interested in learning if the per-acre income on a 

 certain type of farm employing good (recommended by an agricultural 

 experiment station, for example) farming practices is greater, on the 

 average, than that for farmers not following those practices. He takes 

 a sample of n farms on which the recommended practices are employed 

 and calculates the mean per-acre income. The same is done for a 

 comparable random sample of farms on which these recommendations 

 are not followed. Some measurement of the consistency of income on 

 each of the two groups of farms also would be needed. If the newer 

 practices are worth recommending to replace those currently in use, 

 they must produce a new population of per-acre incomes with a larger 

 mean, a smaller variance, or both. To obtain information on these 

 points, the economist must have adequate information regarding the 

 manner in which sample means are distributed; that is, where their 

 region of concentration will be, and how they will tend to be dispersed 

 about that region of concentration. Hence, the first objective of this 

 chapter will be to provide that sort of information about x's drawn 

 from the same normal population of numerical measurements — such 

 as per-acre incomes. 



Figure 6.21 presents the graphs of the frequency distributions shown 

 in Table 6.21. The larger curve is for the near-normal parent popula- 

 tion of X's, while the smaller curve is taken as a good approximation 

 to the distribution of the population of x's obtained from samples of 

 ten observations taken from the population of X's. 



It appears from Figure 6.21 that the two frequency distributions are 

 much alike in general form, and seem to be approximately normal 

 about the same mean. The major difference lies in the fact that the 

 Xi exhibit much less variability than the X's of the population from 

 which the samples were taken. This is to be expected because one 

 important reason for combining a number of individual X's into one 

 sample is to achieve a smoothing out of the individual differences 

 among those X's. 



It should be noted from the bottom of Table 6.21 that the mean of 

 the X{ is 59.98, which is quite near to 60, the size of this population 

 mean, ft. Also, the standard deviation of the 648 sample means is 

 3.14, which is a bit less than one-third of a. As a matter of fact, 3.14 



