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Chapter 23: Problem 6

The data set \(A=\left\\{x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right\\}\) has a mean of \(\bar{x}\) and a standard deviation of \(\sigma .\) The data set \(B\) is \(\left\\{k x_{1}, k x_{2}, k x_{3}, \ldots, k x_{n}\right\\}\), the data set \(C\) is \(\left\\{x_{1}+k, x_{2}+k, x_{3}+k, \ldots, x_{n}+k\right\\}\) where \(k\) is a constant. (a) State the mean of set \(B\). (b) State the mean of set \(C\). (c) State the standard deviation of set \(B\). (d) State the standard deviation of set \(C\).

Short Answer

Expert verified
Answer: The mean of data set B is \(k \times \bar{x}\) and the standard deviation is \(|k| \times \sigma\). The mean of data set C is \(\bar{x} + k\) and the standard deviation is \(\sigma\).

Step by step solution

01

(a) Mean of set B

To find the mean of the data set B, we can use the property of mean: If we multiply each value of the given data set by a constant k, the mean also gets multiplied by the same constant k. So, we can write: Mean of set B = \(k \times \bar{x}\)
02

(b) Mean of set C

To find the mean of the data set C, we can use another property of mean: If a constant, k, is added to all values of a given data set, the mean of the new data set is increased by the same constant, k. So, we can write: Mean of set C = \(\bar{x} + k\)
03

(c) Standard deviation of set B

To find the standard deviation of set B, we can use this property of standard deviation: If each value of the data set is multiplied by a constant k, the standard deviation also gets multiplied by the absolute value of k. So, we can write: Standard deviation of set B = \(|k| \times \sigma\)
04

(d) Standard deviation of set C

To find the standard deviation of set C, we can use another property of standard deviation: If a constant, k, is added to all values of the data set, the standard deviation remains unchanged. So, the standard deviation of set C is the same as the standard deviation of set A. Thus, we can write: Standard deviation of set C = \(\sigma\)

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