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, 4, 8, 10, 13. (a) use the defining formula, the computation formula, or a calculator to compute s. (round your answer to four decimal places.) (b) multiply each data value by 5 to obtain the new data set 85, 20, 40, 50, 65. compute s. (round your answer to four decimal places.) (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 smaller. 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 as large. (d) you recorded the weekly distances you bicycled in miles and computed the standard deviation to be s = 2.9 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? s = km

Explanation:

Step1: Calculate mean of original data set

The original data set is \(17,4,8,10,13\). The mean \(\bar{x}=\frac{17 + 4+8+10+13}{5}=\frac{52}{5} = 10.4\).

Step2: Calculate squared - deviations for original data set

\((17 - 10.4)^2=6.6^2 = 43.56\), \((4 - 10.4)^2=(- 6.4)^2 = 40.96\), \((8 - 10.4)^2=(-2.4)^2 = 5.76\), \((10 - 10.4)^2=(-0.4)^2 = 0.16\), \((13 - 10.4)^2=2.6^2 = 6.76\).

Step3: Calculate variance for original data set

The variance \(s^{2}=\frac{43.56 + 40.96+5.76 + 0.16+6.76}{5 - 1}=\frac{97.2}{4}=24.3\).

Step4: Calculate standard - deviation for original data set

The standard deviation \(s=\sqrt{24.3}\approx4.9295\).

Step5: Calculate mean of new data set

The new data set is \(85,20,40,50,65\). The mean \(\bar{y}=\frac{85 + 20+40+50+65}{5}=\frac{260}{5}=52\).

Step6: Calculate squared - deviations for new data set

\((85 - 52)^2=33^2 = 1089\), \((20 - 52)^2=(-32)^2 = 1024\), \((40 - 52)^2=(-12)^2 = 144\), \((50 - 52)^2=(-2)^2 = 4\), \((65 - 52)^2=13^2 = 169\).

Step7: Calculate variance for new data set

The variance \(s^{2}=\frac{1089+1024 + 144+4+169}{5 - 1}=\frac{2430}{4}=607.5\).

Step8: Calculate standard - deviation for new data set

The standard deviation \(s=\sqrt{607.5}\approx24.6479\).

Step9: Analyze the change in standard deviation

When each data - value is multiplied by \(c = 5\), the original standard deviation \(s_1\approx4.9295\) and the new standard deviation \(s_2\approx24.6479\), and \(s_2 = 5\times s_1\). In general, multiplying each data value by the same constant \(c\) results in the standard deviation being \(|c|\) times as large.

Step10: Convert standard deviation from miles to kilometers

Given \(s = 2.9\) miles and \(1\) mile \(=1.6\) kilometers. The standard deviation in kilometers is \(s=2.9\times1.6 = 4.64\) kilometers. And we don't need to redo all the calculations because changing the units is equivalent to multiplying each data - value by a constant conversion factor.

Answer:

(a) \(4.9295\)
(b) \(24.6479\)
(c) Multiplying each data value by the same constant \(c\) results in the standard deviation being \(|c|\) times as large.
(d) No; \(4.64\)