Statistics: Width of a Confidence Interval

[kingwinner]

Homework Statement

http://www.geocities.com/asdfasdf23135/stat15.JPG

I am OK with part a, but I am having some troubles with part b.

Homework Equations

Width of a Confidence Interval

The Attempt at a Solution

Attempt for part b:
http://www.geocities.com/asdfasdf23135/stat16.JPG
note: P(T>t_(n1+n2-2),alpha/2)=alpha/2 where T~t distribution with n1+n2-2 degrees of freedom.

Now, as n1 increases and n2 increases,
(i) t_(n1+n2-2),alpha/2 gets smaller
(ii) denominator gets larger
(iii) the ∑ terms gets larger because the upper indices of summation are n1 and n2, respectively

(i) and (ii) push towards a narrower confidence interval, but (iii) pushes towards a wider confidence interval. How can we determine the ultimate result?

Any help is greatly appreciated!

 

[kingwinner]

I think the intuitive answer is that the CI will be "narrower", but how can I prove this more rigorously? My method above doesn't seem to work...

 

[Billy Bob]

On your handwritten attempt for part b, use the second to last line instead of the last line. It shows that Sp is just a weighted average of Sx and Sy and therefore should not change much, since the true variances are assumed equal. For your problem, Sx=1.8 and Sy=2.6, so you should assume they are the same and just compute 2t*sqrt{Sp*(etc)} and see that the interval is narrower. Technically Sp might be slightly more or slightly less (and if you compute Sp with Sx=1.8 and Sy=2.6 and then again with interchanged 1.8 and 2.6 you should get an example of both possibilities). The change in t has much more of an effect than any slight change in Sp.

 

[kingwinner]

"Sp should not change much"

Why?? As the sample sizes increase, wouldn't Sx and Sy change?

Thanks!

 

[Billy Bob]

As the sample sizes increase, wouldn't Sx and Sy change?

Even if the sample size stays the same, Sx and Sy probably would change with every experiment.

But you are using them to estimate the true sigma, which by assumption is the same for both Duracell and Energizer.

And Sp is of the form (a*Sx + b*Sy)/(a+b), in other words just a weighted average of these two.

Question (b) really doesn't seem to be posed as a deep question. In fact, the wording of question (b) suggests that you are supposed to assume that the sample means and sample standard deviations are the same as in (a), but the sample sizes are now different.