Answered: Consider the following. 27, 23, 3, 25,…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Consider the following.

27, 23, 3, 25, 24, 9

Compute the population standard deviation of the numbers. (Round your answer to two decimal place.)

(a) Double each of your original numbers and compute the standard deviation of this new population. (Round your answer to two decimal place.)

(b) Use the results of part (a) and inductive reasoning to state what happens to the standard deviation of a population when each data item is multiplied by a positive constant k.

(a)The standard deviation is multiplied by k.

(b)The standard deviation is unchanged.

(c)The standard deviation is divided by k.

(d) The standard deviation is multiplied by -k.

The standard deviation is k.

Expert Solution

Step 1

a) $The mean of 27, 23, 3, 25, 24, 9 is = \frac{27 + 23 + 3 + 25 + 24 + 9}{6} = 18.5$

Then we make a table

data	data-mean	(data - mean)²
27	8.5	72.25
23	4.5	20.25
3	-15.5	240.25
25	6.5	42.25
24	5.5	30.25
9	-9.5	90.25

$Then we add the last column : \sum_{i} {(x_{i} - \bar{X})}^{2} = 495.5 So the population standard deviation is = \sqrt{\frac{\sum_{i} {(x_{i} - \bar{X})}^{2}}{n}} = \sqrt{\frac{495.5}{6}} = 9.087537253 \approx 9.09$

Step 2

b) The new numbers are 54,46, 6, 50, 48, 18 (double of original numbers)

$The mean of 54, 46, 6, 50, 48, 18 is = \frac{54 + 46 + 6 + 50 + 48 + 18}{6} = 37$

Then we make a table

data	data-mean	(data - mean)²
54	17	289
46	9	81
6	-31	961
50	13	169
48	11	121
18	-19	361

$Then we add the last column : \sum_{i} {(x_{i} - \bar{X})}^{2} = 1982 So the population standard deviation is = \sqrt{\frac{\sum_{i} {(x_{i} - \bar{X})}^{2}}{n}} = \sqrt{\frac{1982}{6}} = 18.175074507 \approx 18.18$

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