Data Analysis Alıştırma Soruları — GRE General Test — Sayfa 3

Soru 41Soru →

The table below shows the frequency distribution of customer satisfaction ratings collected by a service center over a 50-day monitoring period.

Rating Interval	Frequency (Days)
1–5	8
6–10	12
11–15	20
16–20	10

Based on the table, what percentage of the days had a customer satisfaction rating of 11 or higher?

20%

40%

50%

60%

75%

Cevabı ve açıklamayı göster

Cevap: 60%

Cevap

60%

To find the percentage of days with a customer satisfaction rating of 11 or higher, add the frequencies of the intervals '11–15' (20 days) and '16–20' (10 days) to get 30 days. The total number of days across all intervals is 8 + 12 + 20 + 10 = 50 days. The percentage is (30 / 50) × 100% = 60%.

Adım Adım Çözüm

1

Identify the relevant rating intervals.

The intervals corresponding to a rating of 11 or higher are '11–15' and '16–20'.

Ratings of 11 or higher include all data points within these two top intervals.

2

Sum the frequencies for the relevant intervals.

Frequency = 20 + 10 = 30 days.

Combining the frequencies gives the total number of favorable outcomes.

3

Calculate the total number of days across all intervals.

Total days = 8 + 12 + 20 + 10 = 50 days.

The total frequency represents the denominator for the percentage calculation.

4

Compute the percentage.

(30 / 50) × 100% = 60%.

Dividing the target frequency by the total frequency and multiplying by 100 yields the required percentage.

Anahtar Kavram

Frequency Table Percentages and Grouped Data Cutoffs

Tahmini Süre:45s

Soru 42Soru →

Dataset $X$ consists of 2 numbers, each equal to 10. Dataset $Y$ consists of 8 numbers, each equal to 25. If Dataset $X$ and Dataset $Y$ are combined to form a single dataset of 10 numbers, what is the standard deviation of the combined dataset?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The standard deviation of the combined dataset is 6.

To find the standard deviation of the combined dataset, first calculate the combined mean:

\frac{2(10) + 8(25)}{10} = 22

. Next, compute the variance by finding the average of the squared deviations from 22:

\frac{2(10 - 22)^2 + 8(25 - 22)^2}{10} = \frac{2(144) + 8(9)}{10} = \frac{360}{10} = 36

. Taking the square root of the variance yields a standard deviation of

\sqrt{36} = 6

.

Adım Adım Çözüm

1

Calculate the mean of the combined 10-number dataset.

The combined mean is 22.

The mean of the combined dataset is needed to compute individual deviations.

2

Compute the squared deviation of each data point from the combined mean and average them to determine the variance.

The variance is 36.

Variance is defined as the arithmetic mean of the squared deviations from the mean.

3

Calculate the square root of the variance to find the standard deviation.

The standard deviation is 6.

Standard deviation is the non-negative square root of variance.

Anahtar Kavram

Standard deviation of a combined dataset

Soru 43Soru →

A dataset of 25 student test scores has a mean of $70$ , a standard deviation of $8$ , and an interquartile range ( $\text{IQR}$ ) of $12$ . A instructor creates a modified dataset by multiplying each original test score by $1.5$ and then adding $10$ to the result. What is the standard deviation of the modified dataset?

8

12

18

22

28

Cevabı ve açıklamayı göster

Cevap: $12$

Cevap

12

For any dataset, if each value

x

is transformed into

a \cdot x + b

, where

a

and

b

are constants, the new standard deviation is given by

|a| \cdot \sigma_{\text{old}}

. In this problem,

a = 1.5

and

b = 10

. Therefore, the new standard deviation is

1.5 \times 8 = 12

. The addition of

10

affects measures of center (such as the mean and median) but has no effect on measures of dispersion (such as standard deviation, IQR, and range).

Adım Adım Çözüm

1

Analyze the effect of multiplying each data point by a constant on standard deviation.

Multiplying every value in a dataset by a constant

c = 1.5

multiplies the standard deviation

\sigma

by

|c|

. The new intermediate standard deviation becomes

1.5 \times 8 = 12

.

Standard deviation measures dispersion around the mean, so scaling all data points by a factor scales the distance between each point and the mean by that same factor.

2

Analyze the effect of adding a constant to each data point on standard deviation.

Adding a constant

k = 10

to every value shifts the entire distribution without changing the distances between data points or the relative spread around the mean. Thus, standard deviation remains

12

.

Adding a constant shifts both the individual values and the mean by the same amount, leaving

(x_i - \bar{x})

unchanged.

Anahtar Kavram

Linear Transformations on Measures of Dispersion

Soru 44Soru →

A researcher recorded seven daily temperature readings (in degrees Celsius): $14, 5, 22, 9, 25, 2,$ and $17$ . What is the median of these seven temperature readings?

9

14

13.5

17

25

Cevabı ve açıklamayı göster

Cevap: $14$

Cevap

The median of the temperature readings is

14

.

To find the median of a set of numbers, first arrange them in ascending order:

2, 5, 9, 14, 17, 22, 25

. Since there are

7

numbers, the median is the middle value in the

4\text{th}

position, which is

14

.

Adım Adım Çözüm

1

Arrange the given dataset in ascending numerical order.

The ordered list of

7

temperatures is:

2, 5, 9, 14, 17, 22, 25

.

To calculate the median of a dataset, the values must first be ordered from smallest to largest.

2

Identify the middle position for an odd number of data values (

n = 7

).

The position of the median is

\frac{7 + 1}{2} = 4\text{th}

position.

For a dataset with

n

elements where

n

is odd, the median is located at position

\frac{n+1}{2}

.

3

Extract the value located at the

4\text{th}

position in the ordered list.

The

4\text{th}

term is

14

.

The fourth number in

2, 5, 9, 14, 17, 22, 25

is

14

.

Anahtar Kavram

Median of a Dataset

Tahmini Süre:45s

Soru 45Soru →

A logistics company recorded the daily delivery processing times (in minutes) for a warehouse over a given period, forming Dataset $X$ . Dataset $X$ has a range of $40$ minutes, an interquartile range ( $\text{IQR}$ ) of $15$ minutes, and a standard deviation of $8.5$ minutes. A new dataset, Dataset $Y$ , is created by transforming each processing time $x$ in Dataset $X$ according to the formula $y = 1.5x - 10$ . Which of the following statements regarding the measures of dispersion for Dataset $Y$ must be true? Select all such statements.

Geçerli olan tümünü seçin

The range of Dataset

Y

is

60

minutes.

The standard deviation of Dataset

Y

is

12.75

minutes.

The interquartile range of Dataset

Y

is

12.5

minutes.

The standard deviation of Dataset

Y

is

2.75

minutes.

The interquartile range of Dataset

Y

is equal to the interquartile range of Dataset

X

.

Cevabı ve açıklamayı göster

Cevap: The range of Dataset $Y$ is $60$ minutes.; The standard deviation of Dataset $Y$ is $12.75$ minutes.

Cevap

The statements asserting that the range of Dataset Y is 60 minutes and that the standard deviation of Dataset Y is 12.75 minutes are both correct.

Under a transformation of the form

y = ax + b

(where

a > 0

), any measure of dispersion

D

transforms according to

D_Y = a \cdot D_X

. The constant

b

does not affect spread. Therefore, the range becomes

1.5 \times 40 = 60

minutes and the standard deviation becomes

1.5 \times 8.5 = 12.75

minutes.

Adım Adım Çözüm

1

Recall the effect of a linear transformation

y = ax + b

on measures of dispersion.

Measures of dispersion (range, IQR, standard deviation) are scaled by

|a|

and are completely unaffected by the constant addition/subtraction

b

.

Adding or subtracting a constant shifts all data points by the exact same amount without altering the relative distances between data points.

2

Calculate the range for Dataset Y.

\text{Range}_Y = 1.5 \times \text{Range}_X = 1.5 \times 40 = 60

minutes.

The multiplicative factor is

a = 1.5

.

3

Calculate the standard deviation for Dataset Y.

S_Y = 1.5 \times S_X = 1.5 \times 8.5 = 12.75

minutes.

The standard deviation scales proportionally by

1.5

.

4

Calculate the interquartile range (IQR) for Dataset Y.

\text{IQR}_Y = 1.5 \times \text{IQR}_X = 1.5 \times 15 = 22.5

minutes.

The IQR also scales proportionally by

1.5

.

Anahtar Kavram

Linear Transformations of Dispersion Measures

Soru 46Soru →

A customer service representative resolved the following number of support tickets over five consecutive days: $18$ , $24$ , $15$ , $31$ , and $22$ . What is the arithmetic mean of the number of tickets resolved per day by the representative?

Cevabı ve açıklamayı göster

Cevap: 22

Cevap

The arithmetic mean of the number of tickets resolved per day is

22

.

The mean of a data set is calculated by taking the sum of all values and dividing by the total number of items. The sum of the numbers is

18 + 24 + 15 + 31 + 22 = 110

. Dividing

110

by

5

yields

22

.

Adım Adım Çözüm

1

Sum all data values in the set

18 + 24 + 15 + 31 + 22 = 110

To find the mean, the first step is to calculate the total sum of all observations.

2

Divide the total sum by the total number of values

\frac{110}{5} = 22

The arithmetic mean is defined as the sum of the values divided by the count of the values.

Anahtar Kavram

Arithmetic Mean

Soru 47Soru →

A researcher recorded the annual rainfall totals (in inches) for a specific region over a 20-year period, forming Dataset $P$ . Dataset $P$ has a range of $18$ inches and an interquartile range (IQR) of $8$ inches. A second dataset, Dataset $Q$ , is created by multiplying each rainfall total in Dataset $P$ by $1.2$ and then subtracting $3$ inches. What is the sum of the range and the interquartile range (IQR) of Dataset $Q$ ?

25.2

26.0

28.2

31.2

37.2

Cevabı ve açıklamayı göster

Cevap: $31.2$

Cevap

31.2

For any linear transformation of the form

Y = aX + b

, measures of spread such as range and interquartile range (IQR) are multiplied by

|a|

, while the constant term

b

has no effect. Multiplying the original range (

18

) and IQR (

8

) by

1.2

yields a new range of

21.6

and a new IQR of

9.6

. Summing these values gives

21.6 + 9.6 = 31.2

.

Adım Adım Çözüm

1

Determine the impact of a linear transformation

Y = aX + b

on measures of dispersion.

Measures of dispersion (Range, IQR, Standard Deviation) scale by

|a|

and are unaffected by the additive constant

b

.

Adding or subtracting a constant shifts all data points by the same amount, leaving the distance between points unchanged, whereas multiplying by a factor scales all distances between points.

2

Calculate the range of Dataset

Q

.

\text{Range}_Q = 1.2 \times \text{Range}_P = 1.2 \times 18 = 21.6

The range scales by the multiplier

1.2

and is not affected by subtracting

3

.

3

Calculate the interquartile range (IQR) of Dataset

Q

.

\text{IQR}_Q = 1.2 \times \text{IQR}_P = 1.2 \times 8 = 9.6

The IQR scales by the multiplier

1.2

and is not affected by subtracting

3

.

4

Compute the sum of the range and the IQR of Dataset

Q

.

\text{Sum} = 21.6 + 9.6 = 31.2

Adding the newly calculated Range and IQR gives the required total spread measure.

Anahtar Kavram

Linear Transformations on Dispersion Metrics

Tahmini Süre:1m 30s

Soru 48Soru →

In a meteorological study, the daily peak wind speeds (in kilometers per hour) recorded over 7 consecutive days were $14, 18, 21, 24, 27, 31,$ and $37$ . As part of a data calibration process, each recorded wind speed is increased by $15\%$ and then increased by an additional constant of $5\text{ km/h}$ . What is the range, in kilometers per hour, of the calibrated daily peak wind speeds?

Cevabı ve açıklamayı göster

Cevap: 26.45

Cevap

The range of the calibrated daily peak wind speeds is 26.45 km/h.

For any dataset transformed linearly according to

y = a \cdot x + b

where

a > 0

, the range of the transformed dataset is given by

\text{Range}(Y) = a \cdot \text{Range}(X)

. Here, increasing each value by

15\%

corresponds to

a = 1 + 0.15 = 1.15

, and adding

5

corresponds to

b = 5

. The original maximum is

37

and the original minimum is

14

, giving an original range of

37 - 14 = 23

. Therefore, the new range is

1.15 \times 23 = 26.45\text{ km/h}

. The constant shift of

5\text{ km/h}

affects the center (mean, median) but has zero effect on the dispersion.

Adım Adım Çözüm

1

Determine the range of the original set of wind speeds.

Original Range = 37 - 14 = 23 km/h.

The range is defined as the difference between the maximum and minimum values in a dataset.

2

Apply the properties of linear transformations to measures of dispersion.

The transformation is y = 1.15x + 5. The range is scaled by 1.15 and unaffected by the addition of 5.

Adding a constant shift to all data points shifts the entire distribution without changing the spread (dispersion), whereas multiplying all data points by a constant factor 'a' scales all measures of dispersion by |a|.

3

Compute the calibrated range.

Calibrated Range = 1.15 * 23 = 26.45 km/h.

Multiplying the original range of 23 by the scale factor of 1.15 gives the exact range of the transformed dataset.

Anahtar Kavram

Effect of linear transformations on measures of dispersion (range, standard deviation, IQR)

Tahmini Süre:1m 30s

Soru 49Soru →

A high school basketball team scored the following points in five consecutive games: $4$ , $6$ , $6$ , $7$ , and $12$ . Which of the following statements regarding the measures of central tendency for these scores must be true? Select all such statements.

Geçerli olan tümünü seçin

The mean score is equal to

7

.

The median score is equal to the mode score.

The median score is equal to

7

.

The mode score is equal to

6

.

The mean score is less than the median score.

Cevabı ve açıklamayı göster

Cevap: The mean score is equal to $7$ .; The median score is equal to the mode score.; The mode score is equal to $6$ .

Cevap

The statements asserting that the mean score is equal to 7, the median score is equal to the mode score, and the mode score is equal to 6 are all true.

For the dataset

4, 6, 6, 7, 12

, the mean is

\frac{35}{5} = 7

, the median (middle score) is

6

, and the mode (most frequent score) is

6

. Consequently, the mean equals

7

, the median and mode are equal (

6 = 6

), and the mode is

6

.

Adım Adım Çözüm

1

Calculate the mean of the dataset.

Mean =

\frac{4 + 6 + 6 + 7 + 12}{5} = \frac{35}{5} = 7

.

The mean is calculated by dividing the sum of all data points by the total count.

2

Determine the median of the dataset.

Median =

6

.

The data is already arranged in ascending order:

4, 6, 6, 7, 12

. The 3rd value (middle position) is

6

.

3

Determine the mode of the dataset.

Mode =

6

.

The value

6

occurs twice, whereas all other values occur only once.

4

Evaluate each given statement against the calculated metrics.

Mean =

7

, Median =

6

, Mode =

6

. Thus: Mean (

7

) is true; Median = Mode (

6 = 6

) is true; Mode =

6

is true.

Direct comparison verifies which statements hold true.

Anahtar Kavram

Calculating and comparing mean, median, and mode for a discrete set of numerical data.

Tahmini Süre:1m 0s

Soru 50Soru →

A company has two sales divisions, Division X and Division Y. Division X has 12 representatives with an average monthly sales volume of $\$14,000$ . Division Y has 18 representatives with an average monthly sales volume of $\$24,000$ . If 2 representatives leave Division Y who together accounted for $\$82,000$ in monthly sales, what is the new combined average monthly sales volume, in dollars, for all remaining representatives in both divisions?

Cevabı ve açıklamayı göster

Cevap: 18500

Cevap

The new combined average monthly sales volume for all remaining representatives is $18,500.

To find the combined mean, calculate the sum of all remaining values across both groups (

518,000

) and divide by the total number of remaining items (

28

), yielding

18,500

.

Adım Adım Çözüm

1

Find total monthly sales for Division X

12 \times 14,000 = 168,000

The total value of a group is the product of its count and its arithmetic mean.

2

Find total monthly sales for Division Y before representatives left

18 \times 24,000 = 432,000

Multiply the number of representatives in Division Y by their average sales volume.

3

Determine the remaining total sales in Division Y

432,000 - 82,000 = 350,000

Subtract the sales volume of the 2 departing representatives from Division Y's original total.

4

Calculate the total combined sales for both divisions

168,000 + 350,000 = 518,000

Add the total sales of Division X and the remaining sales of Division Y.

5

Calculate total number of remaining representatives across both divisions

12 + (18 - 2) = 28

Division X retains 12 representatives, while Division Y retains 16 representatives.

6

Compute the combined mean sales volume

\frac{518,000}{28} = 18,500

Divide the total combined sales volume by the total number of remaining representatives.

Anahtar Kavram

Weighted Mean and Combining Group Data

Soru 51Soru →

Dataset $S$ consists of 11 distinct positive integers arranged in increasing order, with a median of 50, an interquartile range of 20, and a standard deviation of $\sigma$ . A new dataset $S'$ is created by adding 10 to each of the 5 integers in $S$ that are strictly greater than 50, while leaving the remaining 6 integers unchanged. Which of the following statements about dataset $S'$ must be true? Select all such statements.

Geçerli olan tümünü seçin

The interquartile range of dataset

S'

is 30.

The range of dataset

S'

is 10 greater than the range of dataset

S

.

The standard deviation of dataset

S'

is strictly greater than

\sigma

.

The median of dataset

S'

is 60.

The interquartile range of dataset

S'

remains 20.

Cevabı ve açıklamayı göster

Cevap: The interquartile range of dataset $S'$ is 30.; The range of dataset $S'$ is 10 greater than the range of dataset $S$ .; The standard deviation of dataset $S'$ is strictly greater than $\sigma$ .

Cevap

The statements asserting that the interquartile range of dataset

S'

is 30, that the range of dataset

S'

is 10 greater than the range of dataset

S

, and that the standard deviation of dataset

S'

is strictly greater than

\sigma

are all correct.

In an ordered dataset of 11 elements, the median is the 6th element,

Q_1

is the 3rd element, and

Q_3

is the 9th element. When 10 is added only to the elements strictly above the median (elements 7 through 11):

Q_3

increases by 10 while

Q_1

is unchanged, making the new IQR equal to

20 + 10 = 30

. The maximum value increases by 10 while the minimum value remains unchanged, increasing the range by 10. Shifting values in the upper tail further right increases the overall distance of data points from the mean, causing the standard deviation to strictly increase.

Adım Adım Çözüm

1

Analyze the position of quartiles and median in a dataset of 11 ordered values.

For 11 ordered values

x_1 < x_2 < \dots < x_{11}

, the median is

x_6 = 50

,

Q_1 = x_3

, and

Q_3 = x_9

. The lower 6 elements (

x_1

through

x_6

) are unchanged. The upper 5 elements (

x_7

through

x_{11}

) each increase by 10.

Determining which specific data positions change allows us to evaluate median, IQR, and range.

2

Calculate the new interquartile range and range.

Q_{1,\text{new}} = x_3

,

Q_{3,\text{new}} = x_9 + 10

. Thus

\text{IQR}_{\text{new}} = (x_9 + 10) - x_3 = \text{IQR}_{\text{old}} + 10 = 20 + 10 = 30

. The maximum element

x_{11}

increases by 10 while

x_1

is unchanged, so

\text{Range}_{\text{new}} = (x_{11} + 10) - x_1 = \text{Range}_{\text{old}} + 10

.

Interquartile range is

Q_3 - Q_1

and range is

\text{maximum} - \text{minimum}

.

3

Evaluate the effect on the median and standard deviation.

The median remains

x_6 = 50

. Moving the upper values further away from the center increases the overall spread around the mean, which strictly increases the standard deviation beyond

\sigma

.

Standard deviation measures the average spread of values from the mean.

Anahtar Kavram

Effect of asymmetric data shifts on measures of central tendency and dispersion

Soru 52Soru →

Dataset $X$ consists of $50$ distinct real numbers with range $R$ , interquartile range $Q$ , and standard deviation $s$ , where $R > Q > s > 0$ . A new dataset, Dataset $Y$ , is formed by applying the transformation $y = -3x - 7$ to each data point $x$ in Dataset $X$ . If $R_Y$ , $Q_Y$ , and $s_Y$ represent the range, interquartile range, and standard deviation of Dataset $Y$ , respectively, which of the following expressions represents the sum $R_Y + Q_Y + s_Y$ ?

3(R + Q + s)

3(R + Q + s) - 21

3(R + Q + s) - 7

-3(R + Q + s) - 7

-3(R + Q + s)

Cevabı ve açıklamayı göster

Cevap: $3(R + Q + s)$

Cevap

The sum of the dispersion metrics for Dataset

Y

is

3(R + Q + s)

.

Under a linear transformation

y = ax + b

, all measures of dispersion (range, interquartile range, standard deviation) scale by

|a|

and are unaffected by

b

. With

a = -3

and

b = -7

, each measure of dispersion is multiplied by

|-3| = 3

. Therefore,

R_Y = 3R

,

Q_Y = 3Q

, and

s_Y = 3s

, making their sum

3(R + Q + s)

.

Adım Adım Çözüm

1

Identify the effect of linear transformations on measures of dispersion.

For any linear transformation of data

y = ax + b

, measures of dispersion (Range, Interquartile Range, Standard Deviation) scale by the absolute value of the multiplicative constant,

|a|

, and are completely unaffected by the constant addition or subtraction

b

.

Measures of dispersion quantify spread and distances between data points, which shift uniformly when a constant is added but stretch by

|a|

when scaled.

2

Calculate individual dispersion measures for Dataset

Y

.

R_Y = |-3| R = 3R

,

Q_Y = |-3| Q = 3Q

, and

s_Y = |-3| s = 3s

.

The multiplicative factor is

a = -3

, so

|a| = |-3| = 3

. The constant shift

b = -7

has zero effect on spread.

3

Sum the three dispersion measures for Dataset

Y

.

R_Y + Q_Y + s_Y = 3R + 3Q + 3s = 3(R + Q + s)

.

Factoring out

3

yields the simplified combined expression.

Anahtar Kavram

Linear Transformations on Dispersion Metrics

Tahmini Süre:2m 0s

Soru 53Soru →

Dataset $X$ consists of $80$ numerical observations with a standard deviation of $s$ ( $s > 0$ ) and an interquartile range of $I$ ( $I > 0$ ). A new dataset, Dataset $Y$ , is created by transforming each observation $x$ in Dataset $X$ using the linear formula $y = -4x + 25$ . Which of the following correctly gives the standard deviation and the interquartile range of Dataset $Y$ in terms of $s$ and $I$ ?

Standard deviation:

4s

; Interquartile range:

4I

Standard deviation:

-4s + 25

; Interquartile range:

-4I + 25

Standard deviation:

4s + 25

; Interquartile range:

4I + 25

Standard deviation:

-4s

; Interquartile range:

-4I

Standard deviation:

16s

; Interquartile range:

16I

Cevabı ve açıklamayı göster

Cevap: Standard deviation: $4s$ ; Interquartile range: $4I$

Cevap

The standard deviation of Dataset

Y

is

4s

and the interquartile range is

4I

.

For any linear transformation of the form

y = ax + b

, the measures of dispersion (such as standard deviation, interquartile range, and range) scale by

|a|

and are unaffected by the additive constant

b

. Here

a = -4

and

b = 25

, so both standard deviation and IQR scale by

|-4| = 4

, resulting in

4s

and

4I

.

Adım Adım Çözüm

1

Analyze the impact of adding a constant to data values.

Adding

+25

to each data value shifts the position of the data points along the number line, but the relative distances between data points remain constant. Thus, constant addition has zero effect on standard deviation or interquartile range.

Measures of dispersion measure the spread of data around a central value, which is invariant under pure horizontal translations.

2

Analyze the impact of multiplying data values by a scalar factor.

Multiplying each value by

k = -4

expands the distances between points by a factor of

|k| = |-4| = 4

.

Standard deviation and IQR are defined as non-negative distance quantities; scaling data by

k

scales dispersion by

|k|

.

3

Combine the scale and shift transformations.

New Standard Deviation =

|-4| \times s = 4s

, and New IQR =

|-4| \times I = 4I

.

Applying the transformation

y = ax + b

transforms standard deviation

\sigma_y = |a|\sigma_x

and

\text{IQR}_y = |a|\text{IQR}_x

.

Anahtar Kavram

Effect of Linear Transformations on Measures of Dispersion

Soru 54Soru →

A dataset consists of $9$ distinct positive integers $a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9$ listed in strictly increasing order. The mean of the $9$ integers is $28$ , and the median is $24$ . A $10\text{th}$ positive integer $x$ , where $x > a_9$ , is added to the dataset, causing the new mean of the $10$ integers to become $31$ . If $L$ represents the minimum possible value of $a_9$ , what is the value of $x - L$ ?

2

6

21

37

58

Cevabı ve açıklamayı göster

Cevap: $21$

Cevap

The value of

x - L

is

21

.

The sum of the original

9

elements is

9 \times 28 = 252

, and the sum of the

10

elements is

10 \times 31 = 310

, giving

x = 58

. To minimize

a_9

(

L

), the sum of all other elements must be maximized. The median

a_5 = 24

. The maximum possible values for the first four distinct elements below

24

are

20, 21, 22, 23

(summing to

86

). To minimize

a_9

, the upper four elements must be consecutive integers

(a_9 - 3, a_9 - 2, a_9 - 1, a_9)

. Setting the total sum equation

86 + 24 + (4a_9 - 6) = 252

yields

4a_9 = 148

, so

L = 37

. Consequently,

x - L = 58 - 37 = 21

.

Adım Adım Çözüm

1

Calculate the sum of the original 9 integers and determine the value of the 10th integer

x

.

The sum of the original

9

integers is

9 \times 28 = 252

. The sum of the

10

integers after adding

x

is

10 \times 31 = 310

. Therefore,

x = 310 - 252 = 58

.

The sum of a set of numbers equals the number of elements multiplied by the mean.

2

Identify the median of the ordered 9-element set.

In an ordered set of

9

elements, the median is the

5\text{th}

element, so

a_5 = 24

.

For an odd number of ordered elements, the median is the exact middle element.

3

Maximize the sum of the first 4 elements

a_1, a_2, a_3, a_4

to minimize

a_9

.

Since all integers are distinct and strictly increasing,

a_4 < 24

. To maximize

a_1 + a_2 + a_3 + a_4

, choose

a_4 = 23, a_3 = 22, a_2 = 21, a_1 = 20

. Their sum is

20 + 21 + 22 + 23 = 86

.

Maximizing lower elements leaves the smallest possible remainder of the total sum for the upper elements.

4

Express

a_6, a_7, a_8

in terms of

a_9

to minimize

a_9

.

To make

a_9

as small as possible,

a_6, a_7, a_8

should be as large as possible relative to

a_9

, meaning they are consecutive integers:

a_8 = a_9 - 1

,

a_7 = a_9 - 2

,

a_6 = a_9 - 3

.

Making elements above the median consecutive integers directly below

a_9

minimizes

a_9

for a fixed sum.

5

Set up the sum equation for the 9 elements to solve for

L = \text{min}(a_9)

.

(a_1 + a_2 + a_3 + a_4) + a_5 + (a_6 + a_7 + a_8 + a_9) = 252 \implies 86 + 24 + (a_9 - 3 + a_9 - 2 + a_9 - 1 + a_9) = 252 \implies 104 + 4a_9 = 252 \implies 4a_9 = 148 \implies a_9 = 37

. Thus,

L = 37

.

Solving the algebraic equation derived from the total sum yields the minimum integer value for

a_9

while satisfying

a_6 = 34 > 24

.

6

Calculate

x - L

.

x - L = 58 - 37 = 21

.

Subtracting the minimum bound

L

from

x

fulfills the target question requirement.

Anahtar Kavram

Optimization of Dataset Values using Central Tendency Constraints

Tahmini Süre:2m 30s

Soru 55Soru →

A department consisting of $5$ employees has a mean monthly sales total of $\$12{}000$ . If a new employee with a monthly sales total of $\$18{}000$ joins the department, what is the new mean monthly sales total, in dollars, for the $6$ employees?

Cevabı ve açıklamayı göster

Cevap: 13000

Cevap

13000

To find the new mean, multiply the initial mean by the initial number of employees to get total sales:

5 \times 12{}000 = 60{}000

. Add the new employee's sales:

60{}000 + 18{}000 = 78{}000

. Finally, divide by the new total number of employees (

6

) to get

\frac{78{}000}{6} = 13{}000

.

Adım Adım Çözüm

1

Find the total sales of the original 5 employees.

5 \times 12{}000 = 60{}000

The sum of values is equal to the mean multiplied by the number of observations.

2

Calculate the total sales for all 6 employees.

60{}000 + 18{}000 = 78{}000

Add the new employee's sales to the initial total.

3

Calculate the new mean sales per employee.

\frac{78{}000}{6} = 13{}000

Divide the combined total sales by the new total number of employees (6).

Anahtar Kavram

Mean of Combined Data Sets

Tahmini Süre:1m 0s

Soru 56Soru →

A dataset consists of $25$ distinct integers arranged in increasing order. The mean of all $25$ integers is $52$ . The mean of the smallest $12$ integers is $30$ , and the mean of the largest $12$ integers is $70$ .

If $5$ additional numbers, each equal to the median of the original dataset, are added to the dataset, what is the mean of the new set of $30$ numbers?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The mean of the new set of 30 numbers is 60.

The sum of all 25 numbers is

25 \times 52 = 1300

. Since the numbers are ordered, the median is the 13th value. The 12 smallest numbers sum to

12 \times 30 = 360

and the 12 largest sum to

12 \times 70 = 840

. The sum of these 24 numbers is

360 + 840 = 1200

, so the 13th number (the median) must be

1300 - 1200 = 100

. Adding 5 copies of 100 increases the sum to

1300 + 500 = 1800

across

30

numbers. The new mean is

\frac{1800}{30} = 60

.

Adım Adım Çözüm

1

Calculate the total sum of the original dataset of 25 numbers.

Sum =

25 \times 52 = 1300

.

The mean multiplied by the number of elements gives the total sum.

2

Calculate the combined sum of the 12 smallest and 12 largest integers.

Sum of 12 smallest =

12 \times 30 = 360

; Sum of 12 largest =

12 \times 70 = 840

; Total =

360 + 840 = 1200

.

The 25 numbers consist of the 12 smallest, the 1 median (13th element), and the 12 largest.

3

Determine the value of the median.

Median =

1300 - 1200 = 100

.

Subtracting the sum of the 24 non-median values from the total sum yields the 13th element, which is the median.

4

Find the sum and count of the modified dataset.

New Sum =

1300 + 5(100) = 1800

; New Count =

25 + 5 = 30

.

Adding 5 numbers each equal to 100 increases the sum by 500 and the count by 5.

5

Calculate the mean of the new dataset.

New Mean =

\frac{1800}{30} = 60

.

Divide the new total sum by the new total count of numbers.

Anahtar Kavram

Relationship between Mean, Median, and Total Sum in Partitioned Datasets

Tahmini Süre:2m 30s

Soru 57Soru →

A software engineer logged the response time, in milliseconds, for seven independent server requests: $8$ , $16$ , $3$ , $21$ , $14$ , $5$ , and $10$ . What is the median response time, in milliseconds, for these seven requests?

8

10

11

14

21

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

The median response time is

10

milliseconds.

To find the median, first order the seven data points in ascending order:

3, 5, 8, 10, 14, 16, 21

. Since there are

7

values, the median is the middle value at position

\frac{7 + 1}{2} = 4

. The fourth value in this ordered list is

10

.

Adım Adım Çözüm

1

Arrange the given dataset in ascending order.

The sorted dataset is

3, 5, 8, 10, 14, 16, 21

.

Finding the median of a numerical dataset requires ordering the values from smallest to largest first.

2

Determine the position of the median element.

For

n = 7

items, the position is

\frac{7 + 1}{2} = 4

th item.

When the number of observations

n

is odd, the median is the exact middle value located at position

\frac{n+1}{2}

.

3

Identify the value at the 4th position.

The 4th value in the sorted list is

10

.

The 4th element in

3, 5, 8, 10, 14, 16, 21

is

10

.

Anahtar Kavram

Median of a finite numerical dataset with an odd count

Tahmini Süre:45s

Soru 58Soru →

Dataset $P$ consists of the five numbers $10, 20, 30, 40,$ and $50$ . Dataset $Q$ is created by replacing the minimum value in Dataset $P$ with $18$ and the maximum value with $42$ , leaving the remaining three numbers unchanged. Which of the following statements correctly compares the mean and standard deviation of Dataset $Q$ to those of Dataset $P$ ?

The mean of Dataset

Q

is equal to the mean of Dataset

P

, and the standard deviation of Dataset

Q

is less than the standard deviation of Dataset

P

.

The mean of Dataset

Q

is equal to the mean of Dataset

P

, and the standard deviation of Dataset

Q

is greater than the standard deviation of Dataset

P

.

The mean of Dataset

Q

is equal to the mean of Dataset

P

, and the standard deviation of Dataset

Q

is equal to the standard deviation of Dataset

P

.

The mean of Dataset

Q

is greater than the mean of Dataset

P

, and the standard deviation of Dataset

Q

is less than the standard deviation of Dataset

P

.

The mean of Dataset

Q

is less than the mean of Dataset

P

, and the standard deviation of Dataset

Q

is greater than the standard deviation of Dataset

P

.

Cevabı ve açıklamayı göster

Cevap: The mean of Dataset $Q$ is equal to the mean of Dataset $P$ , and the standard deviation of Dataset $Q$ is less than the standard deviation of Dataset $P$ .

Cevap

The mean of Dataset

Q

is equal to the mean of Dataset

P

, and the standard deviation of Dataset

Q

is less than the standard deviation of Dataset

P

.

The mean of both datasets is 30 because the decrease of 8 from 50 balances the increase of 8 to 10. Standard deviation quantifies how far data points deviate from the mean. Because 18 and 42 are closer to the mean of 30 than 10 and 50 are, the spread of Dataset Q around the mean is strictly smaller, making its standard deviation smaller.

Adım Adım Çözüm

1

Calculate the mean of Dataset

P

.

\text{Mean of } P = \frac{10 + 20 + 30 + 40 + 50}{5} = \frac{150}{5} = 30

.

To find the baseline central tendency before the dataset values are modified.

2

Calculate the mean of Dataset

Q

.

\text{Mean of } Q = \frac{18 + 20 + 30 + 40 + 42}{5} = \frac{150}{5} = 30

.

Replacing

10

with

18

(+8) and

50

with

42

(-8) results in a net change of zero to the sum, so the mean remains unchanged.

3

Compare the dispersion of Dataset

Q

relative to Dataset

P

.

In Dataset

P

, the squared deviations of the modified points from the mean are

(10 - 30)^2 = 400

and

(50 - 30)^2 = 400

. In Dataset

Q

, the squared deviations of these points are

(18 - 30)^2 = 144

and

(42 - 30)^2 = 144

.

Standard deviation measures the average distance of data points from the mean. Since the outer values in Dataset

Q

are closer to the mean than in Dataset

P

, the overall dispersion and standard deviation decrease.

Anahtar Kavram

Standard deviation measures the spread of data points around their mean; bringing extreme values closer to the mean reduces the standard deviation.

Soru 59Soru →

Dataset $W$ consists of $15$ distinct real numbers with mean $\mu$ , standard deviation $\sigma > 0$ , interquartile range $I$ , and range $R$ . A 16th numerical value equal to the mean $\mu$ is added to dataset $W$ to form a new dataset $W'$ . Which of the following statements MUST be true regarding dataset $W'$ compared to dataset $W$ ? Select all that apply.

Geçerli olan tümünü seçin

The mean of dataset

W'

is equal to the mean of dataset

W

.

The standard deviation of dataset

W'

is strictly less than the standard deviation of dataset

W

.

The range of dataset

W'

is equal to the range of dataset

W

.

The interquartile range of dataset

W'

must be strictly greater than the interquartile range of dataset

W

.

The standard deviation of dataset

W'

is strictly greater than the standard deviation of dataset

W

.

Cevabı ve açıklamayı göster

Cevap: The mean of dataset $W'$ is equal to the mean of dataset $W$ .; The standard deviation of dataset $W'$ is strictly less than the standard deviation of dataset $W$ .; The range of dataset $W'$ is equal to the range of dataset $W$ .

Cevap

The statements asserting that the mean of dataset

W'

equals the mean of dataset

W

, the standard deviation of dataset

W'

is strictly less than that of dataset

W

, and the range of dataset

W'

equals the range of dataset

W

must all be true.

Adding an element equal to the mean leaves the total sum of squared deviations from the mean unchanged while increasing the sample size by 1. Consequently, the mean remains unchanged, the standard deviation decreases by a factor of

\sqrt{15/16}

, and because the mean lies strictly inside the range of distinct values, the minimum and maximum remain unchanged, preserving the range.

Adım Adım Çözüm

1

Analyze the impact on the mean when adding

x_{16} = \mu

.

The sum of elements in

W'

is

15\mu + \mu = 16\mu

. The new mean is

\frac{16\mu}{16} = \mu

.

Adding a value equal to the mean preserves the mean value.

2

Analyze the impact on the range.

Because all 15 elements are distinct real numbers,

\min(W) < \mu < \max(W)

. Adding

\mu

does not change the minimum or maximum values, so

\text{Range}(W') = \max(W) - \min(W) = R

.

The range depends solely on the maximum and minimum elements.

3

Analyze the impact on the standard deviation.

The sum of squared deviations for

W'

is

\sum_{i=1}^{16} (x_i - \mu)^2 = \sum_{i=1}^{15} (x_i - \mu)^2 + (\mu - \mu)^2 = 15\sigma^2

. The new variance is

\sigma'^2 = \frac{15\sigma^2}{16}

, so

\sigma' = \sigma \sqrt{\frac{15}{16}} < \sigma

.

Increasing the count

n

without increasing the total squared deviation reduces overall dispersion around the mean.

Anahtar Kavram

Effect of adding central summary values on measures of dispersion and central tendency.

Soru 60Soru →

Dataset $S$ consists of $100$ distinct positive numbers. The 25th percentile of dataset $S$ is $40$ , the median is $60$ , and the 75th percentile is $80$ . A new dataset $T$ is formed by multiplying every number in dataset $S$ that is strictly greater than the 80th percentile by $2$ , while keeping all other numbers unchanged. Which of the following statistics MUST be identical for dataset $S$ and dataset $T$ ?

The mean

The standard deviation

The interquartile range

The range

The 90th percentile

Cevabı ve açıklamayı göster

Cevap: The interquartile range

Cevap

The interquartile range

The interquartile range is the difference between the 75th percentile (

Q_3

) and the 25th percentile (

Q_1

). Because only values strictly greater than the 80th percentile are scaled, the values defining

Q_1

and

Q_3

are unaffected. Therefore,

Q_1 = 40

and

Q_3 = 80

remain the same in both datasets, making the interquartile range (

80 - 40 = 40

) identical.

Adım Adım Çözüm

1

Identify which values in the dataset are modified by the transformation

Only numbers strictly greater than the 80th percentile are multiplied by 2. All numbers at or below the 80th percentile remain unchanged.

The transformation criteria specifies that values below or equal to the 80th percentile threshold undergo no change.

2

Evaluate the effect on the 25th percentile (

Q_1

) and 75th percentile (

Q_3

)

Since

Q_1

(25th percentile) and

Q_3

(75th percentile) are position metrics located at or below the 80th percentile mark, neither

Q_1

nor

Q_3

changes in value.

Modifying elements only above the 80th percentile leaves the ordering and exact values of elements up to the 80th percentile completely intact.

3

Determine the impact on the Interquartile Range (IQR)

The interquartile range is defined as

IQR = Q_3 - Q_1

. Because both

Q_1

and

Q_3

remain constant,

IQR

is unchanged.

A metric defined solely by unchanged percentile boundaries must itself remain constant.

Anahtar Kavram

Effect of Upper-Tail Data Transformations on Measures of Position and Dispersion

Tahmini Süre:2m 0s

Data Analysis

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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