A quality control manager records the durability ratings, on a scale from 1 to 50, for 8 r… — GRE General Test

A quality control manager records the durability ratings, on a scale from $1$ to $50$ , for $8$ randomly selected component batches. The ratings are listed below:

34, 18, 42, 27, 18, 39, 45, 25

Two additional component batches with identical ratings, $x$ and $y$ (where $x = y$ ), are included in the dataset. If the median rating of the updated dataset of $10$ batches is equal to the median rating of the original dataset of $8$ batches, what is the value of $x$ ?

A
$22.5$
B
$26.0$
$30.5$ Cevap
D
$31.5$
E
$36.5$

Cevap

The value of

x

30.5

To find the median of the original dataset, the numbers must first be ordered from smallest to largest:

18, 18, 25, 27, 34, 39, 42, 45

. Since there are

8

numbers, the median is the average of the 4th and 5th numbers:

(27 + 34)/2 = 30.5

. When two equal values

x

are added to form a

10

-element dataset, having

x = 30.5

places both new values directly between

27

and

34

. The 5th and 6th terms of the updated set are both

30.5

, making the new median

(30.5 + 30.5)/2 = 30.5

, which matches the original median.

Adım Adım Çözüm

Sort the original dataset of 8 durability ratings in ascending order.

The sorted dataset is

18, 18, 25, 27, 34, 39, 42, 45

Finding the median of a numerical dataset requires arranging the numbers in order from least to greatest.

Calculate the median of the original 8 ratings.

The 4th value is

27

and the 5th value is

34

. The median is

\frac{27 + 34}{2} = 30.5

For a dataset with an even number of elements (

n = 8

), the median is the average of the two middle elements (the 4th and 5th terms).

Analyze how adding two identical values

x

and

y

(

x = y

) affects the median of the 10-element dataset.

Inserting two values equal to

30.5

places them as the 5th and 6th elements in the 10-element sorted array:

18, 18, 25, 27, 30.5, 30.5, 34, 39, 42, 45

. The new median is

\frac{30.5 + 30.5}{2} = 30.5

To keep the median unchanged at

30.5

when adding two identical values, the added values must fall between

27

and

34

and specifically equal

30.5

so that the middle two terms of the 10 values average to

30.5

Anahtar Kavram

Median of a Dataset

Tahmini Süre:1m 30s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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Konular

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