Prove linear function of a variable

[Nyasha]

Homework Statement

In the data table they are noon-hour temperatures of a certain week.

I calculated the mean which is 25 and l also calculated the standard deviation which is 3.74. Now they want me to show that :

if $$y=ax+b$$ then $$\bar{y}=a\bar{x}+b$$ and $$s_{y}= \left| a\right|s_{x}$$

The Attempt at a Solution

I don't know where they get this y from. Can you give me hints on how to solve this question ?

 

[tiny-tim]

Hi Nyasha!

… if $$y=ax+b$$ then $$\bar{y}=a\bar{x}+b$$ and $$s_{y}= \left| a\right|s_{x}$$

I don't know where they get this y from. Can you give me hints on how to solve this question ?

For example, x could be in degrees Celsius, while y could be (9/5)x + 32, which is degrees Fahrenheit.

It's just a change in scale. ​

 

[HallsofIvy]

They are not "getting" y from anywhere. They are defining y to be this linear function of x. The point is to show that if y is a linear function of x then the mean of y is that same linear function of the mean of x and the standard deviation of y is a multiple of the standard deviation of x.

 

[Nyasha]

Okay guys is this correct for the other part which says show that :
$$s_{y}= \left| a\right|s_{x}$$

Attempt to solution:

$$s^2_{y}=(a^2)\cdot(s^2_{x})$$
$$\sqrt{(s^2_{y})}=\sqrt{(a^2)\cdot(s^2_{x})$$
$$s_{y}= \left| a\right|s_{x}$$