Question

In: Statistics and Probability

In this problem, we explore the effect on the standard deviation of multiplying each data value...

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 8, 7, 16, 8, 16.

(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 2 to obtain the new data set 16, 14, 32, 16, 32. 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 remaining the same.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 being |c| times as large.


(d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be s = 2.6 miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calculations?

YesNo    


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

Solutions

Expert Solution

a) mean = 11

variance = So, variance = 21, sd =

b) Now, mean = 22

Now, by the same calculation, variance = 84 = 4 * 21 = 22 * 21

So, sd =

c) Actually, mutiplying 2 makes the variance increased by 22=4 times, which results in making the standard deviation increased by 2 times. Understand that, if we would have multiplied by -2, then the variance would have increased by (-2)2 = 4 times and then standard deviation would increase by 2 times, because standard deviation is never negative.

So, in general, if we multiply each number by c times(c can be positive or negative), then the standard deviation would increase by |c| times, because standard deviation is non-negative.

d) Because 1 mile = 1.6 km, so to get the standard deviation in km, we should multiply each data value by 1.6, and then the standard deviation of those numbers will be 1.6* standard deviation in mile (by the formula we derived in (c))

So, s = 1.6*2.6 km = 4.16km


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