Question

In this problem, we explore the effect on the standard deviation of multiplying each data value in a data set by the same constant. Consider the data set 15, 10, 11, 9, 15. (a) Use the defining formula, the computation formula, or a calculator to compute s.

(Round your answer to one decimal place.) s =

(b) Multiply each data value by 7 to obtain the new data set 105, 70, 77, 63, 105. Compute s. (Round your answer to one decimal place.) s =

(c) Compare the results of parts (a) and (b). In general, how does the standard deviation change if each data value is multiplied by a constant c? Multiplying each data value by the same constant c results in the standard deviation increasing by c units.

Multiplying each data value by the same constant c results in the standard deviation remaining the same.

Multiplying each data value by the same constant c results in the standard deviation being |c| times smaller.

Multiplying each data value by the same constant c results in the standard deviation being |c| times as large.

(d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be s = 3.5 miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calculations? Yes or No

Given 1 mile ≈ 1.6 kilometers, what is the standard deviation in kilometers? (Enter your answer to two decimal places.) s = km

Answer 1

a)

X	(X - X̄)²
15	9.000
10	4.000
11	1.000
9	9.000
15	9.000

sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   2.8

b)

Y	(Y - Ȳ)²
105	441.000
70	196.000
77	49.000
63	441.000
105	441.000

	Y	(Y - Ȳ)²
total sum	420	1568
n	5	5

sample std dev =   √ [ Σ(Y - Ȳ)²/(n-1)] =   19.7990

c)

Multiplying each data value by the same constant c results in the standard deviation being |c| times as large.

d)

Do you need to redo all the calculations? No

s=3.5*1.6 = 5.60km