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Answer:
(a) [tex]\sigma = 7.04[/tex]
(b) [tex]\sigma = 14.1[/tex]
(c) The population standard deviation is multiplied by k
Step-by-step explanation:
Given
[tex]Dataset: 12, 7, 18, 23, 24, 27[/tex]
Solving (a): The population standard deviation
Start by calculating the mean
[tex]\mu = \frac{\sum x}{n}[/tex]
[tex]\mu = \frac{12+7+18+23+24+27}{6}[/tex]
[tex]\mu = \frac{111}{6}[/tex]
[tex]\mu = 18.5[/tex]
The population standard deviation is:
[tex]\sigma = \sqrt{\frac{\sum(x - \mu)^2}{n}}[/tex]
This gives:
[tex]\sigma = \sqrt{\frac{(12-18.5)^2 + (7 - 18.5)^2 + (18-18.5)^2 + (23-18.5)^2 + (24 - 18.5)^2 + (27 - 18.5)^2}{6}}[/tex]
[tex]\sigma = \sqrt{\frac{297.5}{6}}[/tex]
[tex]\sigma = \sqrt{49.5833}[/tex]
[tex]\sigma = 7.04[/tex]
Solving (b): Double the dataset and calculate the new population standard deviation
The new dataset is:
[tex]Dataset: 24, 14, 36, 46, 48, 54[/tex]
Start by calculating the mean
[tex]\mu = \frac{\sum x}{n}[/tex]
[tex]\mu = \frac{24+ 14+ 36+ 46+ 48+ 54}{6}[/tex]
[tex]\mu = \frac{222}{6}[/tex]
[tex]\mu = 37[/tex]
The population standard deviation is:
[tex]\sigma = \sqrt{\frac{\sum(x - \mu)^2}{n}}[/tex]
This gives:
[tex]\sigma = \sqrt{\frac{(24-37)^2 +(14-37)^2 +(36-37)^2 +(46-37)^2 +(48-37)^2 +(54-37)^2}{6}}[/tex]
[tex]\sigma = \sqrt{\frac{1190}{6}}[/tex]
[tex]\sigma = \sqrt{198.33}[/tex]
[tex]\sigma = 14.1[/tex]
Solving (c): What happens when the dataset is multiplied by k
In (a), we have:
[tex]\sigma = 7.04[/tex]
In (b), when the dataset is doubled,
[tex]\sigma = 14.1[/tex]
This implies that when the dataset is multiplied by k, the population standard deviation will be multiplied by the same factor:
i.e.
[tex]New \sigma = k * \sigma[/tex]