Sec. 6.3 ESTIMATION OF p AND a* 159 



have the mean per cent of protein at or above 16. The rarity of such 

 an occurrence might cause you to doubt the accuracy of the protein 

 analyses and cause you to ask that they be done over. 



PROBLEMS 



1. Given that a certain population of measurements is normally distributed 

 about a mean of 30 and with a standard deviation of 8. If a sample of 16 

 members is to be drawn at random, what is the probability that its mean will 

 be below 28? 



2. Under the conditions of problem 1, what is the probability that a sample 

 of 9 numbers taken from that population will have an arithmetic mean below 

 28? Ans. .23. 



3. Solve problem 2 with the standard deviation changed to 12. 



4. If in some particular area the daily wages of coal miners are normally 

 distributed with /i = $15 and a = $1.50 what is the probability that a representa- 

 tive sample of 25 miners will have an average daily wage below $14.25? 



Ans. .006. 



5. Suppose that a thoroughly tested variety of corn has been found to yield an 

 average of 35 bushels per acre with a standard deviation of 6, and that these yields 

 have a normal frequency distribution. If a random sample of 25 yields for a new 

 variety gives x = 40, show that there is good reason to believe that the yields of 

 the new variety are from a population with a mean higher than the 35 bushels per 

 acre for the population of the older variety. 



6.3 ESTIMATION OF THE UNKNOWN MEAN AND 

 VARIANCE OF A POPULATION FROM THE 

 INFORMATION CONTAINED IN A SAMPLE 



If the parameters ju, and o- are unknown for a particular normal 

 population which is being sampled (and they usually are or there 

 would be no occasion for sampling) it becomes necessary to estimate 

 them from the X's taken in the sample. How should this be accom- 

 plished? Although this really is a mathematical problem whose 

 solution lies beyond the scope of this book, certain desirable re- 

 quirements for sampling estimates of fx and a 2 can be considered. 



First, it seems logical that an acceptable estimate should have a 

 mean equal to the corresponding population parameter after many 

 samples have been taken. Even though only one sample of n measure- 

 ments is to be taken, we usually would like to know that the x and' s 2 

 we shall obtain as estimates of ^ and a 2 are from populations whose 

 means are n and a 2 , respectively. Sampling estimates which satisfy 

 this requirement are called unbiased estimates, as noted in Chapter 5. 



The second — and more important — requirement which we should 

 impose on a sampling estimate is that it be as reliable as possible in 



