Question

use the five numbers 18, 15, 12, 13, and 12 to complete parts a) through e) below.

a) compute the mean and standard deviation of the given set of data.
the mean is \\(\bar{x} = 14\\) and the standard deviation is \\(s = 2.55\\).
(round to two decimal places as needed.)

b) add 25 to each of the numbers in the original set of data and compute the mean and the standard deviation of this new set of data.
the new mean is \\(\bar{x} = 39\\) and the new standard deviation is \\(s = 2.55\\).
(round to two decimal places as needed.)

c) subtract 10 from each of the numbers in the original set of data and compute the mean and the standard deviation of this new set of data.
the new mean is \\(\bar{x} = 4\\) and the new standard deviation is \\(s = 2.55\\).
(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 multiplied, or divided, by n.
b. 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.
c. if each piece of data is increased, or decreased, by n, then the mean is increased, or decreased, 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 by n. that is, adding or subtracting a fixed value to a data set will both increase the standard deviation.
b. if each piece of data is increased, or decreased, by n, then the standard deviation is multiplied, or divided, by n.
c. if each piece of data is increased, or decreased, by n, then the standard deviation is increased, or decreased, by n.
d. the standard deviation remains the same if each piece of data is increased, or decreased, by n.

Explanation:

Analyze the effect of addition and subtraction on the mean

Using the Linear Transformations of Data knowledge point

$$
LATEXBLOCK0
$$

Thus, if each piece of data is increased or decreased by \(n\), the mean is increased or decreased by \(n\). This matches option C.

Analyze the effect of addition and subtraction on the standard deviation

Using the Linear Transformations of Data knowledge point

$$
LATEXBLOCK1
$$

Thus, adding or subtracting a constant to each data point does not change the spread, so the standard deviation remains the same. This matches option D.

Answer:

Question 1

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 multiplied, or divided, by n.
• (B) 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.
• (C) If each piece of data is increased, or decreased, by n, then the mean is increased, or decreased, by n. (Correct answer)
• (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 by n. That is, adding or subtracting a fixed value to a data set will both increase the standard deviation.
• (B) If each piece of data is increased, or decreased, by n, then the standard deviation is multiplied, or divided, by n.
• (C) If each piece of data is increased, or decreased, by n, then the standard deviation is increased, or decreased, by n.
• (D) The standard deviation remains the same if each piece of data is increased, or decreased, by n. (Correct answer)