A survey asked consumers to rate a new service on a scale from 1 to 5. The table below sho… — GMAT

A survey asked consumers to rate a new service on a scale from $1$ to $5$ . The table below shows the frequency of each rating received, where $k$ is a positive integer:

Rating	Frequency
1	4
2	6
3	7
4	$k$
5	5

If the median rating of all responses is equal to the arithmetic mean of all responses, what is the value of $k$ ?

A
$2$
B
$3$
$4$ Answer
D
$5$
E
$6$

Answer

4

The option stating

4

is correct because with

k = 4

, the total number of responses is

22 + 4 = 26

and the sum of all ratings is

62 + 4(4) = 78

. The mean is

\frac{78}{26} = 3

. The median of

26

ordered values is the average of the 13th and 14th values, both of which are

3

. Thus, the mean equals the median.

Step-by-Step Solution

Calculate total number of data points and total sum in terms of

k

Total responses

N = 4 + 6 + 7 + k + 5 = 22 + k

. Total sum

= 1(4) + 2(6) + 3(7) + 4(k) + 5(5) = 4 + 12 + 21 + 4k + 25 = 62 + 4k

The mean of a frequency distribution is the total weighted sum divided by the total frequency.

Determine the median rating

Ratings

1, 2, 3

account for the first

4 + 6 + 7 = 17

responses. For small positive integer values of

k

, the middle position(s) fall within the rating of

3

, so the median rating is

3

Since more than half of the initial

22

data points are

3

or below (specifically positions 11 through 17 are 3s), the median is

3

for

k \le 12

Equate the arithmetic mean to the median and solve for

k

\frac{62 + 4k}{22 + k} = 3 \implies 62 + 4k = 3(22 + k) \implies 62 + 4k = 66 + 3k \implies k = 4.

Setting the mean formula equal to the median value of

3

allows solving for

k

algebraically.

Key Concept

Weighted Mean and Frequency Distribution Median

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic