  ### Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836 ### Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836

#### Solutions

Chapter
Section
Chapter 4, Problem 4.5P
Textbook Problem
75 views

Expert Solution
To determine

To find:

The median, mean, range and standard deviation of 13 Canadian provinces and U.S states in two separate years

### Explanation of Solution

Given:

The following table gives the data of median income for Canadian Provinces and Territories of the year 2000 and 2011.

 Province or Territory 2000 2011 Newfoundland and Labrador 38, 800 67, 200 Prince Edward Island 44, 200 66, 500 Nova Scotia 44, 500 66, 300 New Brunswick 43, 200 63, 930 Quebec 47, 700 68, 170 Ontario 55, 700 73, 290 Manitoba 47, 300 68, 710 Saskatchewan 45, 800 77, 300 Alberta 55, 200 89, 930 British Columbia 49, 100 69, 150 Yukon Columbia 56, 000 90, 090 Northwest Territories 61, 000 105, 560 Nunavut 37, 600 65, 280

The following table gives the data of median income for U.S states of the year 1999 and 2012.

 State 1999 2012 Alabama 36, 213 43, 464 Alaska 51, 509 63, 348 Arkansas 29, 762 39, 018 California 43, 744 57, 020 Connecticut 50, 798 64, 247 Illinois 46, 392 51, 738 Kansas 37, 476 50, 003 Maryland 52, 310 71, 836 Michigan 46, 238 50, 015 New York 40, 058 47, 680 Ohio 39, 617 44, 375 South Dakota 35, 962 49, 415 Texas 38, 978 51, 926

Formula used:

The formula to calculate median for odd number of terms is given by,

Median=(Number of terms+1)2

The formula to calculate range is given by,

Range=Highest valueLowest value

Let the data values be Xi’s.

The formula to calculate mean is given by,

X¯=i=1NXiN

The formula to calculate standard deviation is given by,

s=(XiX¯)2N

Where, N is the population size.

Calculation:

For 13 Canadian provinces and Territories

For the year 2000 - arrange the data in the increasing order.

The data in increasing order is given by,

 S.No Income 1 37, 600 2 38, 800 3 43, 200 4 44, 200 5 44, 500 6 45, 800 7 47, 300 8 47, 700 9 49, 100 10 55, 200 11 55, 700 12 56, 000 13 61, 000

The number of terms is 13, which is odd.

The median for the odd number of terms is given by,

Median=(Number of terms+1)2

Substitute 13 for number of terms in the above mentioned formula,

Median=(13+1)2=142=7

The median income corresponding to 7th number is 47, 300.

The highest income is 61, 000 and the lowest income is 37, 600.

The range is given by,

Range=Highest incomeLowest income

Substitute 61, 000 for highest income and 37, 600 for lowest income in the above mentioned formula,

Range=61,00037,600=23400

The size of the population is 13.

The mean is given by,

X¯=i=1NXiN

Substitute 13 for N, 38, 800 for X1, 44, 200 for X2 and so on in the above mentioned formula,

X¯=38,800+44,200+.............+61,000+37,60013=626,10013=48,161.5 ……(1)

Consider the following table of sum of squares,

 Scores (Xi) (Xi−X¯) (Xi−X¯)2 38, 800 −9,361.5 87637682 44, 200 −3,961.5 15693482 44, 500 −3,661.5 13406582 43, 200 −4,961.5 24616482 47, 700 −461.5 212982.25 55, 700 7, 538.5 56828982 47, 300 −861.5 742182.25 45, 800 −2,361.5 5576682.3 55, 200 7, 038.5 49540482 49, 100 938.5 880782.25 56, 000 7, 838.5 61442082 61, 000 12, 838.5 164827082 37, 600 −10,561.5 111545282 ∑(Xi−X¯)=0 ∑(Xi−X¯)2=592950769

From equation (1), substitute 38,800 for X1 and 48161.5 for X¯ in (X1X¯).

(X1X¯)=(38,80048161.5)(X1X¯)=9,361.5

Square the both sides of the equation.

(X1X¯)2=(9.361.5)2(X1X¯)2=87637682

Proceed in the same manner to calculate (XiX¯)2 for all the 1iN for the rest data and refer table for the rest of the (XiX¯)2 values calculated. Then the value of (XiX¯)2 is calculated as,

(XiX¯)2=87637682+15693482+...........+111545282=592950769 ……(2)

The standard deviation is given by,

s=(XiX¯)2N

From equation (2), substitute 592950769 for (XiX¯)2 and 13 for N in the above mentioned formula,

s=59295076913=45611598=6753.64

Thus, range is 23, 400 and standard deviation is 6753.64.

For 13 Canadian provinces and Territories

For the year 2011 - arrange the data in the increasing order.

The data in increasing order is given by,

 S.No Income 1 63, 930 2 65, 280 3 66, 300 4 66, 500 5 67, 200 6 68, 170 7 68, 710 8 69, 150 9 73, 290 10 77, 300 11 89, 930 12 90, 090 13 105, 560

The number of terms is 13, which is odd.

The median for the odd number of terms is given by,

Median=(Number of terms+1)2

Substitute 13 for number of terms in the above mentioned formula,

Median=(13+1)2=142=7

The median income corresponding to 7th number is 68, 710.

The highest income is 105, 560 and the lowest income is 63.930.

The range is given by,

Range=Highest incomeLowest income

Substitute 105, 560 for highest income and 63, 930 for lowest income in the above mentioned formula,

Range=105,56063,930=41,630

The size of the population is 13.

The mean is given by,

X¯=i=1NXiN

Substitute 13 for N, 67, 200 for X1, 66, 500 for X2 and so on in the above mentioned formula,

X¯=67,200+66,500+............+65,28013=971,41013=74723.8 ……(3)

Consider the following table of sum of squares,

 Scores (Xi) (Xi−X¯) (Xi−X¯)2 67, 200 −7,523.8 56607566 66, 500 −8,223.8 67630886 66, 300 −8,423.8 70960406 63, 930 −10,793.8 116506118 68, 170 −6,553.8 42952294 73, 290 −1,433.8 2055782.4 68, 710 −6,013.8 36165790 77, 300 2, 576.2 6636806.4 89, 930 15, 206.2 231228518 69, 150 −5,573.8 31067246 90, 090 15, 366.2 236120102 105, 560 30, 836.2 950871230 65, 280 −9,443.8 89185358 ∑(Xi−X¯)=0 ∑(Xi−X¯)2=1937988108

From equation (3), substitute 67,200 for X1 and 74723.8 for X¯ in (X1X¯).

(X1X¯)=(67,20074723.8)(X1X¯)=7,523.8

Square the both sides of the equation.

(X1X¯)2=(7.523.8)2(X1X¯)2=56607566

Proceed in the same manner to calculate (XiX¯)2 for all the 1iN for the rest data and refer table for the rest of the (XiX¯)2 values calculated. Then the value of (XiX¯)2 is calculated as,

(XiX¯)2=56607566+67630886+.....................+89185358=1937988108 ……(4)

The standard deviation is given by,

s=(XiX¯)2N

From equation (4), substitute 1937988108 for (XiX¯)2 and 13 for N in the above mentioned formula,

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