Elementary Biometrical Inferences 



523 



from these calculations. For convenience, 

 the 95% ranges for f for various values of p 

 and N are plotted in Figure A-2. For 

 values of N not shown, one can interpolate 

 between curves. Note that if N were in- 

 finitely large, f would equal p and for any 

 given value of p, the range would become 

 wider as N decreased. 



B. Range of Parameters Expected from 

 a Statistic Involving One Variable 

 (Figure A-1B) 



If one had no notion what the parameter 

 for the chance of a successful toss of a 

 penny should be, one could make an in- 

 ference about the p value from the sta- 

 tistics obtained. An estimate of the un- 

 known parameter, p, can be obtained from 

 the statistic f. Suppose that 100 tosses of 

 a penny yield 30 successes. The value 

 f = 0.30 is a single statistic. The single 

 best estimate of p is f. From the single 

 f value, the best estimate is p = 0.30. 

 However, it should not be surprising if p 

 were really 0.31, 0.29, or some other nearby 

 value. What would also be valuable to 

 know is the range of p values likely when 

 f = 0.3 and N = 100. This range can be 

 determined by calculating 



f(l - f) 



N 



which is the standard deviation of f, or s f . 

 The values lying between f — 1.96 s f and 

 f + 1.96 s f make up the 95% confidence 

 interval of p, because 95% of the time we 

 would expect this particular sample to have 

 a p value in this interval. If we say that 

 p cannot be outside this range, we will be 

 wrong only 5% of the time. In the present 

 case, s f is about 0.05 and the 95% con- 

 fidence interval of p is roughly 0.20 to 0.40. 

 If one asserts that p must lie between 0.20 

 and 0.40 he will be wrong only about 5% 

 of the time. By reading upward and then 



to the left, one may use Figure A-2 to 

 determine the 95% confidence intervals of 

 p for different values of f. 



PROBLEMS 



A. 1. You suspect that the sex ratio of the 

 fruit fly Drosophila is 0.5 d" c? and 

 0.5 9 9. Let success be cf • What 

 range of successes might you expect 

 with 95% confidence from an un- 

 biased count of 100 flies? 250 flies? 

 1000 flies? 



What is happening to your con- 

 fidence limits as sample size increases? 

 What does this mean? 



A. 2. You expect to draw a sample in 

 which N = 100. What is the 95% 

 range for f when the hypothesis is 

 p = 0.5? p = 0.3? p = 0.1? How 

 does the range of f change according 

 to the hypothesized p values? 



A. 3. You expect 8 different equally-fre- 

 quent types of gametes to be pro- 

 duced by a certain trihybrid. Only 

 one of these is of interest to you. If 

 you sample 50 gametes, what range, 

 in numbers of these interesting gam- 

 etes, are you likely to obtain? 



A. 4. Under certain conditions, white-eyed 

 Drosophila males do not mate very 

 readily with red -eyed females. If 

 the chance of mating is 10%, about 

 how many opportunities for mating 

 should you provide to be reasonably 

 sure that 5 matings will occur? 



A. 5. A student finds 25 brown-eyed flies 

 among 100. Determine with 95% 

 confidence the true probability of a 

 fly's being brown-eyed. 



A. 6. Using Figure A-2, determine the 95% 

 confidence limits of p when f = 0.60, 

 and N = 100, 250, and 1000. 



A. 7. After meiosis of the genotype Aa Bb 

 in Neurospora you obtain 100 asci. 

 If you assume independent segrega- 

 tion, how many ascospores do you 



