Question

use the five numbers 19, 18, 12, 13, and 13 to complete parts a) through e) below.
the new mean is \\( \bar { x } = \\) 10 and the new standard deviation is \\( s = \\) 3.24.
(round to two decimal places as needed.)
d) what conclusions can you draw about changes in the mean and the standard deviation when the same number is added to or subtracted from each piece of data in a set of data?
draw a conclusion about the change in the mean. choose the correct answer below.
a. if each piece of data is increased, or decreased, by n, then the mean is increased by n. that is, adding or subtracting a fixed value to a data set will both increase the mean.
b. if each piece of data is increased, or decreased, by n, then the mean is increased, or decreased, by n.
c. if each piece of data is increased, or decreased, by n, then the mean is multiplied, or divided, by n.
d. the mean remains the same if each piece of data is increased, or decreased, by n.
draw a conclusion about the change in the standard deviation. choose the correct answer below.
a. if each piece of data is increased, or decreased, by n, then the standard deviation is increased, or decreased, by n.
b. if each piece of data is increased, or decreased, by n, then the standard deviation is increased by n. that is, adding or subtracting a fixed value to a data set will both increase the standard deviation.
c. if each piece of data is increased, or decreased, by n, then the standard deviation is multiplied, or divided, by n.
d. the standard deviation remains the same if each piece of data is increased, or decreased, by n.
e) how will the mean and standard deviation of the numbers 8, 9, 10, 11, 12, 13, 14 differ from the mean and standard deviation of 568, 569, 570, 571, 572, 573, 574? determine the mean and standard deviation of both sets of numbers.
for the first set of numbers, the mean is \\( \bar { x } = \\) 11 and the standard deviation is \\( s = \\) 2.16.
for the second set of numbers, the mean is \\( \bar { x } = \\) 578.14 and the standard deviation is \\( s = \\) 2.16.
(round to two decimal places as needed.)

Explanation:

Step1: Understand the effect of adding/subtracting a fixed value

When a fixed value \(n\) is added or subtracted from each data - point in a set:
The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{k}x_{i}}{k}\). If \(y_{i}=x_{i}+n\) (or \(y_{i}=x_{i}-n\)), then \(\bar{y}=\frac{\sum_{i = 1}^{k}(x_{i}+n)}{k}=\frac{\sum_{i = 1}^{k}x_{i}+kn}{k}=\bar{x}+n\) (or \(\bar{y}=\bar{x}-n\)).
The formula for the standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{k}(x_{i}-\bar{x})^{2}}{k - 1}}\). If \(y_{i}=x_{i}+n\) (or \(y_{i}=x_{i}-n\)), then \(s_{y}=\sqrt{\frac{\sum_{i = 1}^{k}[(x_{i}+n)-(\bar{x}+n)]^{2}}{k - 1}}=\sqrt{\frac{\sum_{i = 1}^{k}(x_{i}-\bar{x})^{2}}{k - 1}}=s_{x}\) (the standard deviation measures the spread of data around the mean. Adding or subtracting a constant from each data - point does not change the spread).

Step2: Analyze each option

• Option A: If each piece of data is increased (or decreased) by \(n\), the mean is increased (or decreased) by \(n\). The standard deviation is not affected by adding or subtracting a constant from each data - value. So, this option is incorrect.
• Option B: If each piece of data is increased (or decreased) by \(n\), the mean is increased (or decreased) by \(n\). The standard deviation remains the same. This option is correct.
• Option C: If each piece of data is increased (or decreased) by \(n\), the mean changes. So, this option is incorrect.
• Option D: If each piece of data is increased (or decreased) by \(n\), the mean changes. So, this option is incorrect.

Answer:

B. If each piece of data is increased, or decreased, by \(n\), then the mean is increased, or decreased, by \(n\). The standard deviation remains the same.