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Tough and Tricky questions: Statistics.

During a behavioral experiment in a psychology class, each student is asked to compute his or her lucky number by raising 7 to the power of the student's favorite day of the week (numbered 1 through 7 for Monday through Sunday respectively), multiplying the result by 3, and adding this to the doubled age of the student in years, rounded to the nearest year. If a class consists of 28 students, what is the probability that the median lucky number in the class will be a non-integer?

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14 Oct 2014, 07:22

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Bunuel wrote:

Tough and Tricky questions: Statistics.

During a behavioral experiment in a psychology class, each student is asked to compute his or her lucky number by raising 7 to the power of the student's favorite day of the week (numbered 1 through 7 for Monday through Sunday respectively), multiplying the result by 3, and adding this to the doubled age of the student in years, rounded to the nearest year. If a class consists of 28 students, what is the probability that the median lucky number in the class will be a non-integer?

(A) 0% (B) 10% (C) 20% (D) 30% (E) 40%

lucky number = 3(7^x) + 2a

where a= age of the student

let 28 students be s1,s2,s3.......s28 and assume that there lucky number follows the same order i.e. s1 has the lowest lucky number and s28 has the highest lucky number.

thus median = (s14+s15)/2

now, this median will not be an integer. if one of s14 ,s15 is even an other one is odd.

now let's see. is it possible to have the lucky number as even number.

lucky number =3(7^x) + 2a. now here x will vary from (1 to 7)

= odd (because 3(7^x) will always be odd) + even (2a will always be even) = odd since it is not possible to have lucky number as an even quantity. thus the probability that the median will be a non- integer = 0%

During a behavioral experiment in a psychology class, each student is asked to compute his or her lucky number by raising 7 to the power of the student's favorite day of the week (numbered 1 through 7 for Monday through Sunday respectively), multiplying the result by 3, and adding this to the doubled age of the student in years, rounded to the nearest year. If a class consists of 28 students, what is the probability that the median lucky number in the class will be a non-integer?

(A) 0% (B) 10% (C) 20% (D) 30% (E) 40%

We need to know 2 things to answer this question.

First, each student's lucky number will ALWAYS be an ODD INTEGER. We know this because... lucky number = 7^(student's favorite day of the week - 1,2,3.. or 7) x 3 + (doubled ages of students in years) In other words, lucky number = (ODD INTEGER x ODD INTEGER) + EVEN INTEGER = ODD INTEGER + EVEN INTEGER = ODD INTEGER

Second, when we have an even number of values (28 values), the MEDIAN equals the average (arithmetic mean) of the two middle-most integers (when all of the integers are arranged in ascending order).

Since all 28 values are guaranteed to be ODD (see point #1 above), then we know that the two middle-most integers will be ODD. So, the median of the 28 values = (some ODD integer + some ODD integer)/2 = (an even integer)/2 = an integer. In other words, the median of the 28 values is GUARANTEED to be an integer.

So, P(the median of the lucky numbers will be a non-integer) = 0%

Answer: A

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Re: During a behavioral experiment in a psychology class, each student is
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21 Feb 2017, 07:04