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Question Calculate the range, variance, and standard deviation for the given samples. a. 34,47,45,40,4834, 47, 45, 40, 48, b. 100,3,9160,16207100, 3, 9160, 16207, c. 100,35,40,6040,447100, 35, 40, 6040, 447. a. The range is □\square.

Studdy Solution

STEP 1

Assumptions
1. The samples provided are: a. 34,47,45,40,4834, 47, 45, 40, 48 b. 100,3,9160,16207100, 3, 9160, 16207 c. 100,35,40,6040,447100, 35, 40, 6040, 447
2. The range is the difference between the maximum and minimum values in the sample.
3. The variance is the average of the squared differences from the mean.
4. The standard deviation is the square root of the variance.
5. The calculations are for sample variance and standard deviation, not population variance and standard deviation.

STEP 2

Calculate the range for sample a.
Find the maximum and minimum values of the sample.
Rangea=max⁡(34,47,45,40,48)−min⁡(34,47,45,40,48)\text{Range}_a = \max(34, 47, 45, 40, 48) - \min(34, 47, 45, 40, 48)

STEP 3

Calculate the range for sample a.
Rangea=48−34\text{Range}_a = 48 - 34

STEP 4

Find the numerical value of the range for sample a.
Rangea=14\text{Range}_a = 14

STEP 5

Calculate the range for sample b.
Find the maximum and minimum values of the sample.
Rangeb=max⁡(100,3,9160,16207)−min⁡(100,3,9160,16207)\text{Range}_b = \max(100, 3, 9160, 16207) - \min(100, 3, 9160, 16207)

STEP 6

Calculate the range for sample b.
Rangeb=16207−3\text{Range}_b = 16207 - 3

STEP 7

Find the numerical value of the range for sample b.
Rangeb=16204\text{Range}_b = 16204

STEP 8

Calculate the range for sample c.
Find the maximum and minimum values of the sample.
Rangec=max⁡(100,35,40,6040,447)−min⁡(100,35,40,6040,447)\text{Range}_c = \max(100, 35, 40, 6040, 447) - \min(100, 35, 40, 6040, 447)

STEP 9

Calculate the range for sample c.
Rangec=6040−35\text{Range}_c = 6040 - 35

STEP 10

Find the numerical value of the range for sample c.
Rangec=6005\text{Range}_c = 6005

STEP 11

Calculate the mean for sample a.
Meana=34+47+45+40+485\text{Mean}_a = \frac{34 + 47 + 45 + 40 + 48}{5}

STEP 12

Find the numerical value of the mean for sample a.
Meana=2145\text{Mean}_a = \frac{214}{5}
Meana=42.8\text{Mean}_a = 42.8

STEP 13

Calculate the squared differences from the mean for sample a.
Squared differencesa=(34−42.8)2+(47−42.8)2+(45−42.8)2+(40−42.8)2+(48−42.8)2\text{Squared differences}_a = (34 - 42.8)^2 + (47 - 42.8)^2 + (45 - 42.8)^2 + (40 - 42.8)^2 + (48 - 42.8)^2

STEP 14

Find the numerical values of the squared differences for sample a.
Squared differencesa=(34−42.8)2+(47−42.8)2+(45−42.8)2+(40−42.8)2+(48−42.8)2\text{Squared differences}_a = (34 - 42.8)^2 + (47 - 42.8)^2 + (45 - 42.8)^2 + (40 - 42.8)^2 + (48 - 42.8)^2
Squared differencesa=(−8.8)2+(4.2)2+(2.2)2+(−2.8)2+(5.2)2\text{Squared differences}_a = (-8.8)^2 + (4.2)^2 + (2.2)^2 + (-2.8)^2 + (5.2)^2
Squared differencesa=77.44+17.64+4.84+7.84+27.04\text{Squared differences}_a = 77.44 + 17.64 + 4.84 + 7.84 + 27.04

STEP 15

Calculate the sum of the squared differences for sample a.
Sum of squared differencesa=77.44+17.64+4.84+7.84+27.04\text{Sum of squared differences}_a = 77.44 + 17.64 + 4.84 + 7.84 + 27.04
Sum of squared differencesa=134.8\text{Sum of squared differences}_a = 134.8

STEP 16

Calculate the variance for sample a.
Since we are dealing with a sample and not a population, we divide by n−1n - 1 where nn is the number of observations in the sample.
Variancea=Sum of squared differencesan−1\text{Variance}_a = \frac{\text{Sum of squared differences}_a}{n - 1}
Variancea=134.85−1\text{Variance}_a = \frac{134.8}{5 - 1}

STEP 17

Find the numerical value of the variance for sample a.
Variancea=134.84\text{Variance}_a = \frac{134.8}{4}
Variancea=33.7\text{Variance}_a = 33.7

STEP 18

Calculate the standard deviation for sample a.
Standard deviationa=Variancea\text{Standard deviation}_a = \sqrt{\text{Variance}_a}
Standard deviationa=33.7\text{Standard deviation}_a = \sqrt{33.7}

STEP 19

Find the numerical value of the standard deviation for sample a.
Standard deviationa=33.7≈5.805\text{Standard deviation}_a = \sqrt{33.7} \approx 5.805
For the sake of brevity, I will now calculate the variance and standard deviation for samples b and c without writing down each individual squared difference as done in steps 13-19. The process is the same: calculate the mean, find the squared differences, sum them, divide by n−1n - 1 to get the variance, and take the square root to get the standard deviation.

STEP 20

Calculate the mean for sample b.
Meanb=100+3+9160+162074\text{Mean}_b = \frac{100 + 3 + 9160 + 16207}{4}

STEP 21

Find the numerical value of the mean for sample b.
Meanb=264704\text{Mean}_b = \frac{26470}{4}
Meanb=6617.5\text{Mean}_b = 6617.5

STEP 22

Calculate the variance for sample b.
Varianceb=(100−6617.5)2+(3−6617.5)2+(9160−6617.5)2+(16207−6617.5)24−1\text{Variance}_b = \frac{(100 - 6617.5)^2 + (3 - 6617.5)^2 + (9160 - 6617.5)^2 + (16207 - 6617.5)^2}{4 - 1}

STEP 23

Find the numerical value of the variance for sample b.
Varianceb=42911281.25+43753821.25+6465062.25+91567321.253\text{Variance}_b = \frac{42911281.25 + 43753821.25 + 6465062.25 + 91567321.25}{3}
Varianceb=184584485.53\text{Variance}_b = \frac{184584485.5}{3}
Varianceb≈61528161.833\text{Variance}_b \approx 61528161.833

STEP 24

Calculate the standard deviation for sample b.
Standard deviationb=Varianceb\text{Standard deviation}_b = \sqrt{\text{Variance}_b}
Standard deviationb=61528161.833\text{Standard deviation}_b = \sqrt{61528161.833}

STEP 25

Find the numerical value of the standard deviation for sample b.
Standard deviationb≈7843.9\text{Standard deviation}_b \approx 7843.9

STEP 26

Calculate the mean for sample c.
Meanc=100+35+40+6040+4475\text{Mean}_c = \frac{100 + 35 + 40 + 6040 + 447}{5}

STEP 27

Find the numerical value of the mean for sample c.
Meanc=66625\text{Mean}_c = \frac{6662}{5}
Meanc=1332.4\text{Mean}_c = 1332.4

STEP 28

Calculate the variance for sample c.
Variancec=(100−1332.4)2+(35−1332.4)2+(40−1332.4)2+(6040−1332.4)2+(447−1332.4)25−1\text{Variance}_c = \frac{(100 - 1332.4)^2 + (35 - 1332.4)^2 + (40 - 1332.4)^2 + (6040 - 1332.4)^2 + (447 - 1332.4)^2}{5 - 1}

STEP 29

Find the numerical value of the variance for sample c.
Variancec=1519617.76+1685322.56+1671164.96+22217657.76+787308.964\text{Variance}_c = \frac{1519617.76 + 1685322.56 + 1671164.96 + 22217657.76 + 787308.96}{4}
Variancec=27602071.044\text{Variance}_c = \frac{27602071.04}{4}
Variancec=6900517.76\text{Variance}_c = 6900517.76

STEP 30

Calculate the standard deviation for sample c.
Standard deviationc=Variancec\text{Standard deviation}_c = \sqrt{\text{Variance}_c}
Standard deviationc=6900517.76\text{Standard deviation}_c = \sqrt{6900517.76}
Standard deviationc≈2626.9\text{Standard deviation}_c \approx 2626.9
The calculated range, variance, and standard deviation for each sample are as follows:
a. Range: 14, Variance: 33.7, Standard Deviation: 5.805 b. Range: 16204, Variance: 61528161.833, Standard Deviation: 7843.9 c. Range: 6005, Variance: 6900517.76, Standard Deviation: 2626.9

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