Sec. 4.3 FRACTION OF X'S WITHIN GIVEN LIMITS 97 



that A = (X - 50) /5. If A = 50, A = 0; and if X = 52, A = 0.40. 

 To answer the specific question asked regarding probabilities it is 

 necessary now to extend somewhat the concept of probability pre- 

 viously employed herein. 



When the possible events correspond to positions along a con- 

 tinuous scale of measurement, the number of possibilities (previously 

 denoted by N) is infinite. Moreover, the likelihood of occurrence 

 changes along the scale. It no longer is useful to ask for the prob- 

 ability that X will have a specific size along this scale on any future 

 trial. Instead, an event E will consist of X lying within certain limits. 

 The probability that a randomly chosen X will fall between the 

 limits X = a to X = b now will be defined to be the proportion of 

 all the X's in the population which are included in that interval. 

 Graphically, this will be the proportion of the whole area under the 

 frequency distribution curve which lies between X = a and X = b. 

 Therefore, in problem 4.32, we need to know what proportion of this 

 normal population lies between A = and A = 0.40. From Figure 

 4.31 it is learned that 50 per cent of this population has values less 

 than and about 66 per cent has values less than 0.40; therefore, 

 about 16 per cent of the numbers in a normal population have A's 

 between and 0.40. It follows that P(50 ^ X ^ 52) = .16. 



It is concluded from the symmetry of the normal curve that 

 P(48 ^ X ^ 50) = .16 also. Furthermore, P(60 ^ X ^ 65) = .025 

 because 2.5 per cent of the A's have sizes within the limits 60 to 65. 



As a final illustration of the use to which Figure 4.31 can be put 

 consider a problem of grading "on the curve." 



Problem 4.33. Given that a large group of grades in psychology conform to 

 a normal distribution with /j. = 75 and a = 7, suppose it is required to put letter 

 grades on these scores in the proportions: 7 A '.20.6:46(7: 20D:7F. What are the 

 numerical limits on each letter grade? 



It is useful first to translate the proportionality above into a dif- 

 ferent form as follows. Starting with the lowest grade, F (which 

 will be represented at the left-hand end of the scale of A) , we have the 

 following facts: 



.07 of the grades are to be F ; 



.27 of the grades are to be D or F; 



.73 of the grades are to be C, D, or F; and 



.93 of the grades are to be B, C, D, or F. 



