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 17, 11, 15, 14, 16.
(a) Use the defining formula, the computation formula, or a calculator to compute s. (Round your answer to one decimal place.)
s = 1(b) Multiply each data value by 4 to obtain the new data set 68, 44, 60, 56, 64. Compute s. (Round your answer to one decimal place.)
s = 2
(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 being |c| times as large. 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 being |c| times smaller. Multiplying each data value by the same constant c results in the standard deviation remaining the same.
(d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be s = 2.1 miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calculations?
Yes No
Given 1 mile ≈ 1.6 kilometers, what is the standard deviation in kilometers? (Enter your answer to two decimal places.)
s = 5 km

Answer 1

Answer:

Step-by-step explanation:

a) Mean = (17 + 11 + 15 + 14 + 16)/5 = 14.6

Standard deviation = √(summation(x - mean)²/n

n = 5

Summation(x - mean)² = (17 - 14.6)^2 + (11 - 14.6)^2 + (15 - 14.6)^2 + (14 - 14.6)^2 + (16 - 14.6)^2 = 21.2

Standard deviation = √(21.2/5 = 2.06

Approximating to 1 decimal place, s = 2

The new data set is

68, 44, 60, 56, 64

Mean = (68 + 44 + 60 + 56 + 64)/5 = 58.4

Summation(x - mean)² = (68 - 58.4)^2 + (44 - 58.4)^2 + (60 - 58.4)^2 + (56 - 58.4)^2 + (64 - 58.4)^2 = 339.2

Standard deviation = √(339.2/5 = 8.24

c) The standard deviation of the new data is 4 times the standard deviation of the previous data

In general, multiplying each data value by the same constant c results in the standard deviation being |c| times as large.

d) s = 2.1 miles

Since 1 mile = 1.6 kilometers, the constant with which we would multiply the given standard deviation is 1.6. Therefore, converting to kilometers, it becomes

1.6 × 2.1 = 3.36 km