Problem Solving - 53

A student worked 20 days. For each of the amount shown (see attached table) in the first row of the table, second row gives the number of days the student earned that amount. Median amount of money earned per day for 20 days is?

A) 96
B) 84
C) 80
D) 70
E) 48

Answer: B

Median day = 20+1)/2 = 10.5 th -- money earned was 84
= Average value of 10th and 11th day in the sequence = Median amount of money
Average value of 10th day = 84
Average value of 11th day = 84
Average value of 10th and 11th day = 84 ans

Data Sufficiency - 50

What is the median number of employees assigned per project for the projects at Company Z?

(1) 25 percent of the projects at Company Z have 4 or more employees assigned to each project.
(2) 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project.

Answer: OA - C

From Statement 1): It is given that 25 percent of the projects at Company Z have 4 or more employees assigned to each project - but we donot know the percentage of projects who have employees less than 4 or in other words we do not have any information about the rest 75% projects ---- hence insufficient

From Statement 2): It is given that 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project - but we donot know the percentage of projects who have employees more than 2 or in other words we do not have any information about the rest 65% projects---- hence insufficient

Taking both the statements together:
25 percent of the projects at Company Z have employees 4 , 5, 6..
35 percent of the projects at Company Z have employees 2, 1, 0
=> 40% of projects have 3 employees = > median value is 3

(1-35)employees -- (36-75)employees -- (76-100)employees
2 or less than 2 ---------3, 3, 3, ------------ 4 or more than 4

Problem Solving - 42

E is a collection of four odd integers and the greatest difference between any two integers in E is 4. The standard deviation of E must be one of how many numbers?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7


Answer: B

Suppose the integers are 1, 3 and 5. Therefore the four integers can be:

1, 5, 5, 5
1, 3, 5, 5
1, 3, 3, 5
1, 1, 5, 5
1, 1, 1, 5
1, 1, 3, 5

Here two pairs have the same standard deviation. thus in all we have four different standard deviations

Problem Solving - 41

If eleven consecutive integers are listed from least to greatest, what is the average (arithmetic mean) of the eleven integers?

(1) The average of the first nine integers is 7.
(2) The average of the last nine integers is 9.

Answer: D

Let the numbers be a, b, c, d, e, f, g, h , i, j, k

i) For odd number of consecutive integers median = mean
ii)We also know that the median is the "middle" number in a group (when arranged in ascending or descending order) consisting of an odd number of numbers

We have to find f

From statement (1): It is given that average of first nine numbers = 7
Hence this implies e = 7 ..since it is given numbers are consecutive hence f = 8
Thus sufficient

From statement (2): It is given that average of last nine numbers = 9
Hence this implies g = 9..since it is given numbers are consecutive hence f = 8
Thus sufficient

Problem Solving - 32

List K consists of 12 consecutive integers, if -4 is the least integer in list K, what is the range of the positive integers in the list K?

A. 5
B. 6
C. 7
D. 11
E. 12

Answer: B

The least number in the list is -4, thus the list is: -4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6, 7
Positive integers in the above list: 1, 2, 3, 4, 5, 6, 7
Therefore the range of the positive integers is 7-1 = 6

Data Sufficiency - 36

If set S consist of the numbers 1, 5, -2, 8, and n, is 0 less than n less than 7 ?

1). the median of the numbers in S is less than 5.
2). the median of the numbers in S is greater than 1

Answer: C

From statement (1): Median will be less than 5 only if n is located below 5
Thus the median will either be 1 if n less than 1 or n if 1 less than n less than 5
Hence in both cases n is less than 5 but it can also be n less than 0 ....insufficient

From statement (2): Median will be greater than 1 only if n is located above 1
Thus median will either be 5 if n greater than 5 or n if 1 less than n less than 5
Hence in both cases n greater than 1 but it can also be n greater than 7 ....insufficient

Taking statements (1) and (2) together: 1 less than n less than 5 which lies within the given interval 0 less than n less than 7 ...thus possible values for n are be 2, 3, or 4

Hence the answer C

Problem Solving - 31

A certain characteristic in a large population has a distribution that is symmetric about the mean m.If 68 percent of the distribution lies within one Standard Deviation d of the mean, what percent of the distribution is less than m+d?
A. 16%
B. 32%
C. 48%
D. 84%
E. 92%

Answer: D
In a normal bell curved distribution, 50% are below the mean and 50% are over it
If 68% are distributed within 1 S.D of the mean then this implies that 34% are 1 S.D above the mean and 34% are 1 S.D below the mean i.e 34% between m and m+d and 34% between m-d and m
The distribution is symmetric about m also => 32/2 = 16% between 0 and m-d and 16% m+d and above.
Hence total that is less than m+d = 100-16 = 84%

OR

Distribution is symmetric around mean => 68/2 = 34% =>(Mean-S.D, Mean) = (Mean, Mean+S.D] = 34% . Thus below Mean+S.D = 50+34 = 84%

NOTE: For a normal bell-curve distribution, the percentage is approx 34% between the mean and 1 S.D. the percentage is approximately 13.6% between 1 SD and 2 SD, the percentage is approximately 2% between 2 S.D and on...

Statistics - Rules and Tips!

1. Mean Average = total of quantities / number of quantities
2. The median is the "middle" number in a group (when arranged in ascending or descending order) consisting of an odd number of numbers, and the average of the two middle numbers if there are an even number of numbers
3. For a set of consecutive integers, the median is the the average of the first and the last integer
4. Mode is the most frequently recurring number/numbers among the given set of numbers. It can be more than one
5. Range is the difference between the largest number and smallest number is a set
6. Calculation of Standard Deviation (SD):

1. Find the mean, , of the values.
2. For each value x_i calculate its deviation () from the mean.
3. Calculate the squares of these deviations.
4. Find the mean of the squared deviations. This quantity is the variance σ².
5. Take the square root of the variance.

7. Variance is the square of the standard deviation
8. SD does not change when the same constant is added or subtracted to all the members of the set
9. If mean = maximum value it means that all values are equal and SD is 0
10. A set of numbers with range of zero means that all of the numbers are the same, hence the dispersion of the numbers from its mean is zero
11. For data with approximately the same mean, the greater the spread, the greater the SD.
12. SD is the square root of the average of the sum of square of the variation from the mean
13. The more uneven members are dispersed around their arithmetic average, the more their SD
14. You only need to know the difference between values and total number of values to compute SD
15. If we know all the numbers of the list, there is a definite SD, regardless of what it is, we can compute it and get an answer – this is helpful for DS questions
16. If the range is 0, then the SD must also be 0, because there is no variance
17. The SD of any list is not dependent on the average, but on the deviation of the numbers from the average. So just by knowing that two lists having different averages doesn't say anything about their standard deviation - different averages can have the same SD
18. The sum of the deviations of the elements from the mean must be 0
19.Closer the more values to the MEAN, lower the SD
20. If Range or SD of a list is 0, then the list will contain all identical elements
21. Standard Deviation is also useful when comparing the spread of two separate data sets that have approximately the same mean. The data set with the smaller Standard Deviation has a narrower spread of measurements around the mean and therefore usually has comparatively fewer high or low values.
In general, the more widely spread the values are, the larger the Standard Deviation is.
22. If you multiply all terms by x then SD =x times old SD and mean = x times old mean
23. For comparing the SD for two sets any information about mean ,median,mode and range are insufficient unless you can determine the individual terms from the given data
24. Symmetric about the mean means that the shape of the distribution on the right and left side of the curve are mirror-images of each other
25. For a given set of consecutive even numbers.. mean = median
26. When you have a set of consecutive numbers (integers, evens, odds, multiples), the mean is equal to the median
27. Good links for Normal Distribution:

http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html

http://davidmlane.com/hyperstat/z_table.html

http://www.integratedlearning.net/gmat/sample.asp

Problem Solving - 15

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%

Answer - A
Since any power of 7 is odd, the product of this power and 3 will always be odd. Adding this odd number to the doubled age of the student (an even number, since it is the product of 2 and some integer) will always yield an odd integer. Therefore, all lucky numbers in the class will be odd.

The results of the experiment will yield a set of 28 odd integers, whose median will be the average of the 14th and 15th greatest integers in the set. Since both of these integers will be odd, their sum will always be even and their average will always be an integer. Therefore, the probability that the median lucky number will be a non-integer is 0%.

Manhattan challenge problem of the week -- June 26

If set S = {7, y, 12, 8, x, 9}, is x + y less than 18?

(1) The range of set S is less than 9.

(2) The average of x and y is less than the average of set S.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer - B

(1) INSUFFICIENT: Statement (1) tells us that the range of S is less than 9. The range of a set is the positive difference between the smallest term and the largest term of the set. In this case, knowing that the range of set S is less than 9, we can answer only MAYBE to the question "Is (x + y) <>

Consider the following two examples:Let x = 7 and y = 7. The range of S is less than 9 and x + y <>

Let x = 10 and y = 10. The range of S is less than 9 and x + y > 18, so we conclude NO.

Because this statement does not allow us to answer definitively Yes or No, it is insufficient.

(2) SUFFICIENT: Statement (2) tells us that the average of x and y is less than the average of the set S. Writing this as an inequality:

(x + y)/2 < (7 + 8 + 9 + 12 + x + y)/6

(x + y)/2 < (36 + x + y)/6

3(x + y) <>

2(x + y) <>

x + y <>

Therefore, statement (2) is SUFFICIENT to determine whether x + y <>