1. The following are the number of hours that 10 police officers have spent being trained in how to handle encounters with people who are mentally ill: 4 17 12 9 6 10 1 5 9 3 Calculate the (a) range, (b) - QAmmunity.org

1. The following are the number of hours that 10 police officers have spent being trained in how to handle encounters with people who are mentally ill: 4 17 12 9 6 10 1 5 9 3 Calculate the (a) range, (b)

asked Mar 15, 2021 109k views

1 Answer

Answer:

Range = 16

$Inter\ Quartile\ Range = 6.75$

Variance = 20.44

$Standard\ Deviation = 4.52$

Explanation:

Given

4, 17, 12, 9, 6, 10, 1, 5, 9, 3

Calculating the range;

Range = Highest - Lowest

From the given data;

Highest = 17 and Lowest = 1

Hence;

Range = 17 - 1

Range = 16

Calculating the Inter-quartile Range

Inter quartile range (IQR) is calculates as thus

IQR = Q_3 - Q_1

Where

Q3 = Upper Quartile and Q1 = Lower Quartile

Start by arranging the data in ascending order

1, 3, 4, 5, 6, 9, 9, 10, 12, 17

N = Number of data; N = 10

---------------------------------------------------------------------------------

Calculating Q3

$Q_3 = (3)/(4)(N+1) th\ item$

Substitute 10 for N

$Q_3 = (3)/(4)(10+1) th\ item$

$Q_3 = (3)/(4)(11) th\ item$

$Q_3 = (33)/(4) th\ item$

$Q_3 = 8.25 th\ item$

Express 8.25 as 8 + 0.25

$Q_3 = (8 + 0.25) th\ item$

$Q_3 = 8th\ item + 0.25 th\ item$

Express 0.25 as fraction

$Q_3 = 8th\ item +(1)/(4) th\ item$

$Q_3 = 8th\ item +(1)/(4) (9th\ item - 8th\ item)$

From the arranged data;

$8th\ item = 10$ and
$9th\ item = 12$

$Q_3 = 8th\ item +(1)/(4) (9th\ item - 8th\ item)$

Q_3 = 10 +(1)/(4) (12 - 10)

Q_3 = 10 +(1)/(4) (2)

Q_3 = 10 +0.5

Q_3 = 10.5

Calculating Q1

$Q_1 = (1)/(4)(N+1) th\ item$

Substitute 10 for N

$Q_1 = (1)/(4)(10+1) th\ item$

$Q_1 = (1)/(4)(11) th\ item$

$Q_1 = (11)/(4) th\ item$

$Q_1 = 2.75 th\ item$

Express 2.75 as 2 + 0.75

$Q_1 = (2 + 0.75) th\ item$

$Q_1 = 2nd\ item + 0.75 th\ item$

Express 0.75 as fraction

$Q_1 = 2nd\ item +(3)/(4) th\ item$

$Q_1 = 2nd\ item +(3)/(4) (3rd\ item - 2nd\ item)$

From the arranged data;

$2nd\ item = 3$ and
$3rd\ item = 4$

Q_1 = 3 +(3)/(4) (4 - 3)

Q_1 = 3 +(3)/(4) (1)

Q_1 = 3 +0.75

Q_1 = 3 .75

---------------------------------------------------------------------------------

Recall that

IQR = Q_3 - Q_1

IQR = 10.5 - 3.75

IQR = 6.75

Calculating Variance

Start by calculating the mean

Mean = (1+3+4+5+6+9+9+10+12+17)/(10)

Mean = (76)/(10)

Mean = 7.6

Subtract the mean from each data, then square the result

(1 - 7.6)^2 = (-6.6)^2 = 43.56

(3 - 7.6)^2 = (-4.6)^2 = 21.16

(4 - 7.6)^2 = (-3.6)^2 = 12.96

(5 - 7.6)^2 = (-2.6)^2 = 6.76

(6 - 7.6)^2 = (-1.6)^2 = 2.56

(9 - 7.6)^2 = (1.4)^2 = 1.96

(9 - 7.6)^2 = (1.4)^2 = 1.96

(10 - 7.6)^2 = (2.4)^2 = 5.76

(12 - 7.6)^2 = (4.4)^2 = 19.36

(17 - 7.6)^2 = (9.4)^2 = 88.36

Sum the result

43.56 + 21.16 + 12.96 + 6.76 + 2.56 + 1.96 + 1.96 + 5.76 + 19.36 + 88.36 = 204.4

Divide by number of observation;

Variance = (204.4)/(10)

Variance = 20.44

Calculating Standard Deviation (SD)

SD = √(Variance)

SD = √(20.44)

SD = 4.52 (Approximated)

Dali answered Mar 19, 2021

by Dali

7.9k points

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