100 BINOMIAL AND NORMAL DISTRIBUTIONS Ch. 4 



less than 50 will favor this decision, that is, it will seem that the residents are 

 against this decision when they actually favor it? Ans. .13. 



3. Suppose that an event E 1 occurs with a relative frequency p = 1/2, and 

 that n random observations are to be made under these conditions. How large 

 must n be before the number of occurrences, r, of E x will fall within one per 

 cent of its mathematical expectation with probability equal to .10? That is, 

 you must choose n so that P(n/2 - ra/200 ^ r ^ n/2 + n/200) = .10. 



4. Suppose that a basketball team has established in previous games that it 

 is safe to assume that the probability on each shot by a team member that a 

 goal will be scored is .35. What is the probability that in a game in which they 

 take 60 shots from the field they will hit less than 18 if the idealized assump- 

 tions just stated are good? Ans. .17. 



5. Suppose that a pair of unbiased dice are to be rolled 50 times. What is 

 the probability that a 6 or a 7 or an 8 will appear on 20 to 25, inclusive, of these 

 throws? 



6. According to certain records the average length of growing season at Man- 

 hattan, Kansas, is 172 days. If the standard deviation about this mean is 13 

 days, and if lengths of growing seasons in this area are normally distributed, 

 what is the probability that the next growing season will be long enough to 

 mature a crop which requires 190 days to complete its development? Ans. .084. 



7. Suppose that when wood blocks of a certain type, 2 by 2 by 8 inches, are 

 tested for strength with the proper engineering equipment, their strengths are 

 normally distributed with mean equal 13,000 pounds and standard deviation 

 equal 3600 pounds. How many blocks out of 100 tested would you expect to 

 have strengths below 6000 pounds? Between 10,000 and 15,000 pounds? 



8. If you are told that the heights of 10,000 college men closely follow a 

 normal distribution with fi = 69 inches and <r = 2.5 inches: 



(a) How many of these men would you expect to be at least 6 feet in height? 



Ans. 1150. 



(b) What range of heights would you expect to include the middle 50 per 

 cent of the men in the group? Ans. 67.3 to 70.7. 



9. Assuming that the wages of certain laborers in the building trades are 

 normally distributed about a mean of $1.80 per hour with a standard deviation 

 of 30 cents: 



(a) What proportion of the laborers receive at least one dollar per hour? 



(b) What range includes the middle two-thirds of these laborer's wages? 



10. Suppose that tests have indicated that certain silk fabrics have breaking 

 strengths which are normally distributed about a mean of 27 pounds, with 

 a = 8; whereas, materials with a mixture of silk and rayon have n = 37 pounds 

 and a = 9. How likely is it that a piece of silk selected at random will be at 

 least as strong as the average for the silk and rayon mixture? How likely is 

 it that a randomly chosen piece of the silk-rayon mixture will be no stronger 

 than the average for silk? Ans. P = .11, .13. 



11. Suppose that the persons whose ACE scores are in Table 2.01 are to be 

 given letter grades on the assumption that these scores are normally distributed 

 with /j. = 95.7 and a = 26.1. If 10 per cent are to get A's and 22 per cent D's, 

 compute the score limits on each letter grade. 



