Statistics Test-3

Given:

• Set P is an arithmetic sequence with common difference(d) = 4 and number of terms(n) = 10
• Let the first term be a.
  • So, the other 9 terms = {a+d, a+2d, ……a+9d}
• (a+9d) and (a+8d) are removed

To Find : Decrease in the average of the set after removal of (a+9d) and (a+8d)

Approach:

1. To calculate the decrease in the average of the sequence, we need to calculate the average of the sequences before and after removal of the terms (a+9d) and (a+8d)
2. Calculating Average of the original set P
    
   • As we know all the terms of the set P in terms of a, we can calculate the sum of all the terms in set P using the sum of an arithmetic sequence formula.
     •   ​Once, we know the sum of an arithmetic sequence P, we can  calculate the average of the arithmetic sequence by dividing the sum by the number of terms(i.e. 10)
3. Calculating Average of set P after removal of (a+9d) and (a+8d)
   • The new sum of the arithmetic sequence can be calculated by subtracting the sum of (a+9d) and (a+8d) from the original sum of the arithmetic sequence
     • The new average can then be calculated by dividing the new sum by the remaining terms in the sequence(i.e. 8)

Working out:

1. Calculating Average of the original set P
   • Sum of the 10

Hence, the average decreased by 4 units.

Answer : C

Given:

• A = Average of {3, 6, 9 . .. 21}
• B = Median of {n, 2n, 3n} where n is a positive integer
  • So, B = 2n
  • Since n is a positive integer, B must be a positive integer as well
• A² - B2 =0  

To Find: n =?

Approach:

1. To find the value of n, we need to draft an equation in terms of n. From the given information, we can write n= B/2  So, to find n, we need to find the value of B
2. We are given a relation between A² and B² . So, if we know the value of A, we can find the value of B
3. We can find the value of A using the given information about A

Working out:

• Rejecting the negative value since we ’ve inferred above that B is a positive integer
• So, B = 12

Looking at the answer choices, we see that the correct answer is Option D

Step 1 &2: Understand Question and Draw Inference

Given:

• Set P = {A, B, C} where A, B and C are distinct positive integers
• Average of Set P =  

To find: Is    divisible by 3?

• If A + B + C divisible by 9?
• Is =   an integer?
• Let A = 9m + a, where a is the remainder that A leaves with 9. So a lies between 0 and 8, inclusive
• Similarly let B = 9n + b
• And let C = 9p + c
• So, the question is to find: Is   an integer?
• Is   an integer?
• Since m + n + p is an integer, the question simplifies to find: is  

Step 3 : Analyze Statement 1 independent

• So, the number 10,000a + 100b +c is divisible by 9
• By applying the divisibility rule of 9, we can say that the sum of the digits of the number 10,000a + 100b +c  is divisible by 9
  • Since a, b and c are each single-digit integers that lie between 0 and 8, inclusive, the number  10,000a + 100b +c is of the form  a00b0c (that is, a is the ten-thousands digit of this number, b is the hundreds digit and c is the units digit).
• So, the sum of digits of 10,000a +100b+c  (=a00b0c) = a+b+c
• So, a+b+c  is divisible by 9
• So,  is an integer  

Sufficient.

Step 4 : Analyze Statement 2 independent

• The product of the range of Set P and the median of Set P is 18.
• (Range of Set P)*(Median of Set P) = 18
  • Since Set P contains only integers, Range of Set P is an integer
  • Since Set P contains an odd number of integers, Median of Set P is equal to the middle integer when A, B and C are arranged in ascending order. Thus, Median of Set P is an integer as well.
• Thus, the product of 2 integers is 18
• The number of ways to express 18 as a product of 2 integers:
  • Case 1: 18 = 18*1
    • Either Range = 18 and Median = 1
      • Not possible –A, B and C are distinct, positive integers. So, minimum value of an integer in Set P is 1. The second integer in Set P cannot also be 1. So, Median cannot be 1
    • Or Range = 1 and Median = 18
      • Not possible –A, B and C are distinct integers. So, the minimum possible value of Range is 2 (happens if A, B and C are consecutive integers)
  • Case 2: 18 = 9*2
    • Either Range = 9 and Median = 2
      • This means, Set P = {1, 2, 10}
      • A + B + C = 1 + 2 +10 = 13, which is not divisible by 9.
      • So, the answer to the question is NO
    • Or Range = 2 and Median = 9
      • This means, Set P = {8, 9, 10}
      • A + B + C = 8 + 9 + 10 = 27, which is divisible by 9
      • So, the answer to the question is YES
  • Case 3: 18 = 6*3
    • Since we ’ve already seen that Case 2 leads to conflicting answers, we need not analyze Case 3.

Thus, Statement 2 is not sufficient to provide a unique answer to the posed question.

Step 5: Analyze Both Statements Together (if needed)

Since we ’ve already arrived at a unique answer in Step 3, this step is not required.
Answer: Option A

Step 1 &2: Understand Question and Draw Inference

Given : A set of n distinct integers, arranged in the order of increasing magnitude
To find: Is Median = Mean?
The median is equal to the mean if:

• Either the given sequence is an Arithmetic progression
• Or, It ’s not an Arithmetic Progression but symmetric about a number.
  • Like the set, {1, 2, 4, 6, 7}. This set is {4 –3, 4 –2, 4, 4 + 2, 4 + 3}, i.e. it is symmetric about 4.So, the sum of the terms = 4*5 And   

Step 3 : Analyze Statement 1 independent

(1) The sum of any 3 successive integers of the set is divisible by 3

• Statement 1 is fulfilled by more than one cases:
• Case 1 : The terms of the set are in arithmetic progression
  • Example: 1, 2, 3, 4, 5
  • In this case, as discussed in Step 1 and 2, Median = Mean (= 3 in the Example above)
  • Case 2 : The terms of the set are not in arithmetic progression
• For example, a set of the form: {3k + 0, 3k + l, 3k + 5, 3k + 12, 3k + 16} where k is an integer
• In this set, 
• But Median = 3k + 5
• So, Median ≠Mean
• Thus, Statement 1 doesn ’t give us a unique answer to the asked question. So, this statement is not sufficient

Step 4 : Analyze Statement 2 independent

(2) The difference between any 2 successive integers of the set is 4

• Note that we are given that the integers are arranged in ascending order.
• So, combining this fact with Statement 2, we can write that the numbers are of the form: {m, m + 4, m + 8, m + 12, . . . , m + (n-1)*4}
• Thus, the given sequence is an Arithmetic Progression.
• Therefore, the median of the sequence will definitely be equal to the mean of the sequence.

So Statement 2 is sufficient.

Step 5: Analyze Both Statements Together (if needed)

Since we ’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B

Step 1 &2: Understand Question and Draw Inference

• Set A has 15 positive integers

To Find: Is Mean (A) = Median (A)

Step 3 : Analyze Statement 1 independent

1. The integers in set A, when arranged in the order of increasing magnitude, are not evenly spaced

• For an evenly spaced set, mean = median
• However, for a non-evenly spaced out, mean may or may not be equal to median. For example:
  • {1, 2, 5, 8, 9}. Here mean = median = 5
  • {1, 3, 6, 8, 22}. Here mean = 8 but median = 6

Insufficient to answer.

Step 4 : Analyze Statement 2 independent

2. If an integer x, which is equal to the mean of set A, is added to the set, the median of the set does not change

• When set A consists of 15 terms. Arranging them in ascending order, we have,
  • Median = 8 term = let ’s say y
• When set A consists of 16 terms, i.e. after the addition of a new term
  • Median = Average of 8 and the 9 term = y(as the median did not change)
  • 8^th term + 9^th term = 2y
• Now, let ’s observe the position of the term y in the new set A
  • If x
  • Then y would become the 9 term of the new set A
• If x = y
  • Then the 8^th term and the 9^th term both will be equal to y
• If x >y
  • Then y would remain the 8 term of the new set A

• So, even after adding of the new term x, y (i.e. the 8 term of the original set A) would either be the 8 term or the 9 term in the new set A
  • Hence, from the equation 8 term + 9 term = 2y, we can say that Either(8 term or 9 term) = y
  • So, 8 term = 9 term = y
• For the above to hold true, following cases are possible:
  • Case-I: mean = median = x .
    • This is true for the case, when x = y
    • If the same number is added, the mean and the median would not change
  • Case-II: When median ≠mean
    • In this case the median ≠mean and the median value is repeated more than once, such that addition of a new number results in the median being the average of the two repeated median values. For example: Set consisting of {3, 3, 3, 9, 9, 9, 9, 9, 9}. Mean = 7 and
      median = 9
      If 7 is added, Set = {3, 3, 3, 7, 9, 9, 9, 9, 9, 9}. Mean = 7 and
      median = (9+9 ) /2 = 9
  • So, the mean = median or the mean ≠median Insufficient to answer

Step 5: Analyze Both Statements Together (if needed)

1. Set A is not evenly spaced
2. If an integer x, which is equal to the mean of set A, is added to the set, the median of the set does not change

Combining the two statements does not give us any extra information to answer the question.
Insufficient to answer.

Answer : E

Given:

• Total Selling Price for 32 items = $28800
• Profit margin = 20%

To Find: Average cost per item

Approach:

1. Average cost per item  ​

• We know that the number of items is 32
• So, to answer the question, we need to know the total cost price of all items

2. We ’re given the total selling price of all items as well as the profit margin. Using these 2 pieces of information together, we can find the total cost price of all 32 items

Working out:

Looking at the answer choices, we see that the correct answer is Option B

Step 1 &2: Understand Question and Draw Inference

• List A has 16 distinct odd integers and 9 distinct even integers
  • Total number of integers in List A = 16 + 9 = 25
• Average of List A = 13.84
  • Sum of all integers in List A = 13.84 * 25 = 346
  • Let the odd integers in List A be D₁ , D₂ , D₃ . . .D₁₆ Let the sum of these integers be D
  • Let the Even integers be E , E , E . . . E . Let the sum of these
  • integers be E.
  • D + E = 346 . . . (1)
• When each odd number is doubled, the new sum of the odd numbers = 2D.

So, in order to answer this question, we need to know the value of D.

Step 3 : Analyze Statement 1 independent

(1) The average (arithmetic mean) of the even integers in List A is 10

Putting (2) in (1), we get the value of D.
So, Statement 1 alone is sufficient.

Step 4 : Analyze Statement 2 independent

(2) Before each odd integer is doubled in magnitude, the smallest odd integer in
List A is 1 and the largest odd integer in List A is 31

• D₁ = 1
• D₁₆ = 31
• Number of odd integers between 1 and 31, inclusive, is 16.
• This means, the 16 odd integers in List A are consecutive odd integers starting from 1 and ending at 31.
• Since we now know the values of the odd integers in List A, we can find their sum D.

So, Statement 2 alone is sufficient.

Step 5: Analyze Both Statements Together (if needed)

Since we get a unique answer in each of Steps 3 and 4, this step is not required

Answer: Option D

Step 1 &2: Understand Question and Draw Inference

• Let the number of bugs the engineer found on day 1 = a
  • Number of bugs he found on day 2 = a-x
  • Number of bugs he found on day 3 = a- x- x = a - 2x
  • Number of bugs he found on day n = a - (n-1)x
• Thus the number of bugs found by the engineer is a decreasing Arithmetic Progression.

To Find: standard deviation of the number of bugs found during the last 7 days

• To find the standard deviation of a set, we need to know the following:
  • Number of data points in the set
    • We are given the number of days = 7
• Distance of each data point of the set from the mean of the set
  • We are given the difference between the bugs found on consecutive days = x
  • If we know the value of x, we can find the distance of each data point from its mean.
• So, we need to know the value of x to calculate the standard deviation

Step 3 : Analyze Statement 1 independent

• The difference between the maximum number of bugs and the minimum number of bugs found by the engineer during the last 7 days is 24.
• Maximum number of bugs found = (Number of bugs found on day 1) = a
• Minimum number of bugs found = (Number of bugs found on day 7) = a –6x
• a - (a - 6x) = 24
  • x = 6

Sufficient to answer

Step 4 : Analyze Statement 2 independent

2. The average (arithmetic mean) number of bugs found by the engineer during the last 7 days is 24

• Total number of bugs found in during the last 7 days = a + (a –x) +(a-2x) +……(a-6x) = 7a –21x
• Average number of bugs found =
• a - 3x = 24

1 equation 2 variables àcannot find a unique value of x.
Insufficient to answer

Step 5: Analyze Both Statements Together (if needed)

Since, we have a unique answer from step- 3, this step is not required.
Answer: A

Step 1 &2: Understand Question and Draw Inference

Given: 4 boys and 5 girls take a test
To find: The Average score of the group

Step 3 : Analyze Statement 1 independent

Statement 1 says that ‘The average score of the boys is 23 points while the average score of the girls is 20 points ’

• Therefore, we will be able to find the value of the Average Score of the group
• Statement 1 is sufficient to answer the question

Step 4 : Analyze Statement 2 independent

Statement 2 says that ‘If one of the girls had scored 6 points more, the average  score of the group would have been 22 ’

Let the girl who scored 6 points be the first girl (in the list of scores). So, her new score = G + 6

So, Statement 2 is sufficient to answer the question

Step 5: Analyze Both Statements Together (if needed)

Since we ’ve already arrived at a unique answer in each of Steps 3 and 4, this step is not required
Answer: Option D

Given:

• Number of athletes = 20
• Table showing range of distances, rounded to the nearest integers, run by the athletes

To Find: The option that cannot be the average distance run by the athl

Approach:

1. We need to look for a value that is outside the range of the average distance run by the 20 athletes.
2. We know that the average distance would lie between the minimum possible average distance ran by the athletes and maximum possible average distance ran by the athletes
   • So, for finding the range of the distances run, we need to find the maximum and minimum average distances run by the athletes.
   • Once we know the maximum and minimum distance run, we can find the range of average distance run by dividing by the number of athletes.
3. Maximum Average Distance
   • The average distance ran by the athletes will be maximum when each athlete runs the maximum distance in a given range
     • For example: In the range 1- 10 kilometers, each athlete should run 10 kilometers
     • Maximum distance run = Sum of (maximum distance run in the range * number of athletes in the range)
     • Maximum Average distance run = 

Working out:

1. Minimum distance run = (4*1) + (5*11) +(6*21) + (3*31) +(2*41) = 4 + 55 +126 +93 +82 = 360
   • Average of minimum distance run = 
2. Maximum distance run = (4 * 10) + ( 5 *20) +(6*30) + (3*40) +(2*42) = 40 + 100 + 180 +120 + 84 = 524 kilometres
   • a. Average of maximum distance run =  
3. Hence the range of average distance run by the athletes can be from 18 to 26.2
4. The only option that is outside the range is option E, 28 kilometres

Answer : E