46. use this data set: 10, 20, 30, 40, 50
a. find the standard deviation.
b. add 5 to each value, and then find the standard deviation.
c. subtract 5 from each value and find the standard deviation.
d. multiply each value by 5 and find the standard deviation.
e. divide each value by 5 and find the standard deviation.
f. generalize the results of parts b through e.
g. compare these results with those in exercise 38 of exercises 3 - 1.
47. the mean deviation is found by using this formula:
mean deviation = \frac{\sum|x - \overline{x}|}{}

Question

46. use this data set: 10, 20, 30, 40, 50
 a. find the standard deviation.
 b. add 5 to each value, and then find the standard deviation.
 c. subtract 5 from each value and find the standard deviation.
 d. multiply each value by 5 and find the standard deviation.
 e. divide each value by 5 and find the standard deviation.
 f. generalize the results of parts b through e.
 g. compare these results with those in exercise 38 of exercises 3 - 1.
 47. the mean deviation is found by using this formula:
 mean deviation = \frac{\sum|x - \overline{x}|}{}

Accepted Answer

# Explanation:
## Step1: Calculate the mean for part a
The mean $\bar{x}=\frac{10 + 20+30+40+50}{5}=\frac{150}{5} = 30$
## Step2: Calculate the variance for part a
$$
\begin{align*}
s^{2}&=\frac{(10 - 30)^{2}+(20 - 30)^{2}+(30 - 30)^{2}+(40 - 30)^{2}+(50 - 30)^{2}}{5-1}\
&=\frac{(- 20)^{2}+(-10)^{2}+0^{2}+10^{2}+20^{2}}{4}\
&=\frac{400 + 100+0+100+400}{4}\
&=\frac{1000}{4}=250
\end{align*}
$$
## Step3: Calculate the standard deviation for part a
$s=\sqrt{250}\approx15.81$

## Step4: For part b: New data set $15,25,35,45,55$
The mean $\bar{x}=\frac{15 + 25+35+45+55}{5}=\frac{175}{5} = 35$
$$
\begin{align*}
s^{2}&=\frac{(15 - 35)^{2}+(25 - 35)^{2}+(35 - 35)^{2}+(45 - 35)^{2}+(55 - 35)^{2}}{5-1}\
&=\frac{(-20)^{2}+(-10)^{2}+0^{2}+10^{2}+20^{2}}{4}\
&=\frac{400+100 + 0+100+400}{4}=250
\end{align*}
$$
$s=\sqrt{250}\approx15.81$

## Step5: For part c: New data set $5,15,25,35,45$
The mean $\bar{x}=\frac{5+15+25+35+45}{5}=\frac{125}{5} = 25$
$$
\begin{align*}
s^{2}&=\frac{(5 - 25)^{2}+(15 - 25)^{2}+(25 - 25)^{2}+(35 - 25)^{2}+(45 - 25)^{2}}{5-1}\
&=\frac{(-20)^{2}+(-10)^{2}+0^{2}+10^{2}+20^{2}}{4}\
&=\frac{400+100+0+100+400}{4}=250
\end{align*}
$$
$s=\sqrt{250}\approx15.81$

## Step6: For part d: New data set $50,100,150,200,250$
The mean $\bar{x}=\frac{50+100+150+200+250}{5}=\frac{750}{5}=150$
$$
\begin{align*}
s^{2}&=\frac{(50 - 150)^{2}+(100 - 150)^{2}+(150 - 150)^{2}+(200 - 150)^{2}+(250 - 150)^{2}}{5-1}\
&=\frac{(-100)^{2}+(-50)^{2}+0^{2}+50^{2}+100^{2}}{4}\
&=\frac{10000 + 2500+0+2500+10000}{4}\
&=\frac{25000}{4}=6250
\end{align*}
$$
$s=\sqrt{6250}\approx79.06$

## Step7: For part e: New data set $2,4,6,8,10$
The mean $\bar{x}=\frac{2 + 4+6+8+10}{5}=\frac{30}{5}=6$
$$
\begin{align*}
s^{2}&=\frac{(2 - 6)^{2}+(4 - 6)^{2}+(6 - 6)^{2}+(8 - 6)^{2}+(10 - 6)^{2}}{5-1}\
&=\frac{(-4)^{2}+(-2)^{2}+0^{2}+2^{2}+4^{2}}{4}\
&=\frac{16+4+0+4+16}{4}=10
\end{align*}
$$
$s=\sqrt{10}\approx3.16$

## Step8: For part f
If $y_{i}=x_{i}+k$ (where $k$ is a constant), the standard deviation $s_y=s_x$. If $y_{i}=kx_{i}$ (where $k
eq0$ is a constant), $s_y = |k|s_x$

# Answer:
a. $s\approx15.81$
b. $s\approx15.81$
c. $s\approx15.81$
d. $s\approx79.06$
e. $s\approx3.16$
f. Adding or subtracting a constant to each data - value does not change the standard deviation. Multiplying each data - value by a non - zero constant $k$ changes the standard deviation by a factor of $|k|$.

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the mean for part a

Step2: Calculate the variance for part a

Step3: Calculate the standard deviation for part a

Step4: For part b: New data set \(15,25,35,45,55\)

Step5: For part c: New data set \(5,15,25,35,45\)

Step6: For part d: New data set \(50,100,150,200,250\)

Step7: For part e: New data set \(2,4,6,8,10\)

Step8: For part f

Answer: