Sec. 5.3 PREDETERMINED HYPOTHESES REGARDING p 135 



place a confidence interval on p. For example, the 95 per cent con- 

 fidence interval could be obtained (from a larger table than 5.21 on 

 page 127) . Such an interval would include p — 3/4, because that hy- 

 pothesis was accepted far above the 5 per cent level, but would also 

 include other possible values of p. If this interval included other 

 defensible hypotheses about p, they also would be acceptable as far 

 as this sample evidence is concerned. Larger samples then could 

 be taken with the hope of so narrowing the confidence interval that 

 only one theoretically defensible hypothesis would be acceptable 

 upon the basis of the sampling evidence. 



PROBLEMS 



1. According to Table 2.61, 50 female and 59 male guinea pigs were born 

 during the period from January to April, inclusive. If these guinea pigs can 

 be considered as a random sample of all guinea pigs as regards the sex ratio, 

 is the observed difference in numbers of each sex sufficient to cause you to 

 reject the hypothesis that the sex ratio actually is 1:1, if you wish to set the 

 probability of committing an error of the first kind at .05? 



2. Use the data for May to August, inclusive, to answer the question of 

 problem 1. Am. No, P( x 2 ^ 1.12) ££ .30. 



3. Solve as in problem 1 for the data for September to December. 



4. Table 2.62 contains data from those guinea pigs which survived long enough 

 to produce 4-day gains. Do the data for January to July, inclusive, indicate that 

 the sex ratio is 1:1 for guinea pigs in that more select population which lives 

 at least 4 days? Am. Yes, P > .53. 



5. Solve as in problem 4 for the data for August to December. 



6. According to genetic theory, if a so-called heterozygous red-eyed fruit fly 

 is mated with a white-eyed fruit fly, one-half the offspring are expected mathe- 

 matically to be white-eyed. The reasoning is analogous to that given earlier 

 for a mating of O and AB blood types. Suppose that among 500 offspring of 

 such fruit flies, 240 are white-eyed. Does the x 2 -test indicate that such a sam- 

 ple result would occur rarely (P < .05) while sampling from a binomial pop- 

 ulation with p = 1/2, or not? Am. No, P = .37. 



7. If you assume (as is reasonable from Figure 5.31) that x 2 must be at least 

 3.8 in problem 6 before the hypothesis that p = 1/2 should be rejected, how 

 small can the number of white-eyed flies be among 500 offspring before that 

 would occur? 



8. Suppose that it were agreed that you should not seriously doubt the 

 hypothesis that p = 1/2 unless x 2 exceeds a value xo 2 which is such that 

 P(x 2 — Xo 2 ) — .01. How small can the number of white-eyed flies among 500 

 become before you would reject the hypothesis that p = 1/2? 



Ans. 221 or 222. 



9. Suppose that a sample of 100 college students showed that 40 opposed a 

 certain proposal regarding student government. Does that result contradict 

 the hypothesis that 48 per cent of the student body oppose the proposed change? 



