Is the Standard Deviation the Same for Two Years?

[Pinto09]

Homework Statement

The electricity usage (KWh) in homes was studied over a period of time.

For Year 1:
Mean = 1130
Std dev. = 196
n = 16

For Year 2:
Mean = 1350
Std dev. = 302
n = 21

a)
Test the hypothesis that the std dev. in year 2 has increased compared to year 1. Remember to state your significance level.

b)
Test the hypothesis that the std dev. in year 1 and 2 are equal. Remember to state your significance level.

Homework Equations

The Attempt at a Solution

I don't know about a) really.

For b) I think the F-distribution could be used, right?

But what significance level should I use? Someone help me get started on this?

 

[Mark44]

I think that the F statistic is the one to use for both tests. My only concern is the relatively small sample sizes for each problem part.

From what you have posted, it looks like it's your choice on the significance level. Pick one, but be sure to state it in your work.

For the first part, the null and alternate hypotheses are these:
H₀: s₁² <= s₂²
H_a: s₁² > s₂²

Because of the > in the alternate hypothesis, you need to use a one-tailed test.

For the second part, the null and alternate hypotheses are these:
H₀: s₁² != s₂² (i.e., they're not equal)
H_a: s₁² = s₂²
For this problem you need to use a two-tailed test.

 

[statdad]

The F distribution can be used, but the hypotheses in Mark44's post should involve $$\sigma_1^2$$ and $$\sigma_2^2$$, and not the sample variances. I think he just made a quick typo.

 

[Pinto09]

All right, thanks!
But do you think I should use alfa = 5%, or is another number better?

 

[Mark44]

As far as I can tell, it's your call on what to choose for alpha.

 

[Pinto09]

Just one last question
Since s(y) > s(x)
where x= year 1 and y= year 2

In a)
should I reject H(0) if

s(x)^2 / s(y)^2 < F n(x)-1, n(y)-1, (a)
or reject if
s(x)^2 / s(y)^2 > F n(x)-1, n(y)-1, (a)?

 

[Mark44]

Draw the curve for each situation, and determine what the critical values are. If your statistic lies in the tail (one-tailed test) or in either tail (two-tailed test), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.