Elementary Statistics: A Step By Step Approach
Elementary Statistics: A Step By Step Approach
10th Edition
ISBN: 9781259755330
Author: Allan G. Bluman
Publisher: McGraw-Hill Education
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Chapter 3.2, Problem 18E

Use the data from Exercises 7, 15, and 17 (unemployment, prisoners, precipitation days) and compare the standard deviation with that obtained by the range rule of thumb (R/4.) Comment on the results.

Traveler Spending The data show the traveler spending in billions of dollars for a recent year for a sample of the states. Find the range, variance, and standard deviation for the data.

20.1 33.5 21.7 58.4 23.2 110.8 30.9 24.0
74.8 60.0            

Source: U.S. Travel Agency.

Laws Passed The data show the number of public laws passed by the U.S. Congress for a sample of recent years. Find the range, variance, and standard deviation for the data.

283 394 383 580 498 460 377 482

Source: Congressional Record.

Annual Precipitation Days The number of annual precipitation days for one-half of the 50 largest U.S. cities is listed below. Find the range, variance, and standard deviation of the data.

135 128 136 78 116 77 111 79 44 97
116 123 88 102 26 82 156 133 107 35
112 98 45 122 125          
Expert Solution & Answer
Check Mark
To determine

The standard deviation by standard formula and by range rule of thumb for the given data and compare the results.

Answer to Problem 18E

The standard deviation for first second and third data is 29.86,91.5 and 34.01 respectively. The standard deviation for first second and third data by range rule of thumb is 22.675,74.25 and 32.5 respectively. The standard deviation by range rule of thumb is an approximate result, it is not appropriate.

Explanation of Solution

Given info:

Refer to Exercise 7,15 and 17.

The data shows the unemployment as given below:

20.1 33.5 21.7 58.4
23.2 110.8 30.9 24.0
74.8 60.0

The data shows the prisoners as given below:

283 394 383 580 498 460 377 482

The data shows the precipitation days as given below:

135 128 136 78 116 77 111 79 44 97
116 123 88 102 26 82 156 133 107 35
112 98 45 122 125

Calculation:

The formula for standard deviation is,

s=(XX¯)2n1 (1)

Construct the table to find the above values.

S. no. Values XX¯ (XX¯)2
1 20.1 25.64 657.4096
2 33.5 12.24 149.8176
3 21.7 24.04 577.9216
4 58.4 12.66 160.2756
5 23.2 22.54 508.0516
6 110.8 65.06 4232.804
7 30.9 14.84 220.2256
8 24.0 21.74 472.6276
9 74.8 29.06 844.4836
10 60.0 14.26 203.3476
Total 457.4 8026.964

The formula to calculate the mean is,

X¯=1nXnn=X1+X2+X3+...+Xnn (2)

Where,

  • X is the number of observers and active watchers.
  • n is the total number of states.

Substitute 110Xn=457.4 and n=10 in equation (2),

Mean=457.410=45.74

Substitute (XX¯)2=8026.964 and n=10 in equation (1),

s=8026.964101=8026.9649=891.884929.86

The standard deviation for the first data is 29.86 .

Construct the table to find the standard deviation for second data.

S. no. Values XX¯ (XX¯)2
1 283 149.125 22238.27
2 394 38.125 1453.516
3 383 49.125 2413.266
4 580 147.875 21867.02
5 498 65.875 4339.516
6 460 27.875 777.0156
7 377 55.125 3038.766
8 482 49.875 2487.516
Total 3457 58614.88

Substitute 18Xn=34.57 and n=8 in equation (2),

Mean=34578=432.125

Substitute (XX¯)2=58614.88 and n=8 in equation (1),

s=58614.8881=58614.887=8373.55491.5

The standard deviation for the second data is 91.5.

Construct the table to find the standard deviation for third data.

S. no. Values XX¯ (XX¯)2
1 135 36.16 1307.55
2 128 29.16 850.31
3 136 37.16 1380.87
4 78 20.84 434.31
5 116 17.16 294.47
6 77 21.84 476.99
7 111 12.16 147.87
8 79 19.84 393.63
9 44 54.84 3007.43
10 97 1.84 3.39
11 116 17.16 294.47
12 123 24.16 583.71
13 88 10.84 117.51
14 102 3.16 9.99
15 26 72.84 5305.67
16 82 16.84 283.59
17 156 57.16 3267.27
18 133 34.16 1166.91
19 107 8.16 66.59
20 35 63.84 4075.55
21 112 13.16 173.19
22 98 0.84 0.71
23 45 53.84 2898.75
24 122 23.16 536.39
25 125 26.16 684.35
Total 2471 27761.36

Substitute 125Xn=2471 and n=25 in equation (2),

Mean=247125=98.84

Substitute (XX¯)2=27761.36 and n=25 in equation (1),

s=27761.36251=27761.3624=1156.72334.01

The standard deviation for the third data is 34.01.

The formula to calculate the standard deviation by range rule is,

sRange4 (3)

Where,

  • The range is the difference between the largest data value and the smallest data value.

For the unemployment data, the smallest data value is 20.1 and the largest data value is 110.8, therefore the difference between them is 90.7.

Substitute 90.7 for range in equation (3),

s90.74=22.675

The standard deviation for the first data is 22.675.

For the prisoner data, the smallest data value is 283 and the largest data value is 580, therefore the difference between them is 297.

Substitute 297 for range in equation (3),

s2974=74.25

The standard deviation for the second data is 74.25.

For the precipitation day data, the smallest data value is 26 and the largest data value is 156, therefore the difference between them is 130.

Substitute 130 for range in equation (3),

s1304=32.5

The standard deviation for the third data is 32.5.

Therefore, standard deviation for first second and third data is 29.86,91.5 and 34.01 respectively. The standard deviation for first second and third data by range rule of thumb is 22.675,74.25 and 32.5 respectively. The standard deviation by range rule of thumb is an approximate result, it is not appropriate.

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Chapter 3 Solutions

Elementary Statistics: A Step By Step Approach

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