According to the National Association of Homebuilders, the average
life expectancy of a dishwasher and a garbage disposal are about the
same: 10 years. Assume that their finding  was based on a sample of
n1=60 dishwashers and n2=40 garbage disposals, and that the
corresponding sample standard deviations were s1=3.0 years and s2=3.7
years. Using the 0.02 level of significance, examine whether the
population standard deviations for the lifetimes of these two types of
appliances could be the same.(Hyp Test, two sample means)

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Request for Question Clarification by jeremymiles-ga on 17 Apr 2003 04:50 PDT

What do you mean when you say "two sample means".  This is comparing
two sample proportions.

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Clarification of Question by wta2k-ga on 17 Apr 2003 05:12 PDT

Right, " or proportions" ^_^

Hi yet again, wta2k!
The proof of every step I will do now is more carefully explained in
my answer to your question #191622. In answering this one, I will
assume that you have understood that question, as the explanation here
is exactly the same. Again, in this question you're not comparing
neither means nor proportions, you're comparing standard deviations of
two groups.

As you should have seen in question #191622, the F-test will compare
variances instead of standard deviations. Of course, it's still useful
for this question, since you can find the variance by squaring the
standard deviation.

Let's call S1 and S2 the sample *variances*, not the sample standard
deviations. So, we have
S1= (3.0)^2 = 9.0
S2= (3.7)^2 = 13.6

We want to see if S2/S1 is significantly different from 1. As we are
checking the null hypothesis that both variances are equal versus the
alternative hypothesis that the veriances are not equal, this will be
a two-tail test. We should reject the null hypothesis if S2/S1 is
"much" greater than 1 or "much" smaller than 1. That is, we have two
rejection areas. The level of significance is 0.02, so we will leave
0.01 on each tail to be the rejection hypothesis. That is, we will
have to find 2 k's (the term 'k' was defined in qeustion #191622). We
have to find k1 such that:
Prob ( S2/S1 > k1 ) = 0.01
and k2 such that
Prob ( S2/S1 < k2 ) = 0.01

We will reject the null hypothesis if the actual value S2/S1 is either
greater than k1 (that is, if it is "sufficiently" greater than 1) or
smaller than k2 (that is, "sufficiently" lower than 1).

It turns out that we don't need to find the value for k2. This is
because we already know the value for S2/S1 and we know that it is
greater than 1. We will obviously not reject it for being sufficiently
smaller than 1. Thus, we find k1 just as we did in the previous
question.

S2/S1 will follow an F distribution with 39 numerator degrees of
freedom (df) and 59 denominator df. Recall that these numbers come
from the number of observations in each group. We now check the table,
at alpha=0.01, to find that the F-value for alpha=0.01 is 2.019.

Since S2/S1 = 13.6 / 9 = 1.5111 < 2.019, we don't reject the null
hypothesis that both population variances are equal. S2 is not
"sufficiently" greater than S1, and it is obviously not smaller than
S1. So we conclude that at the 0.02 level of significance, we can't
reject the hypothesis that both population variances are equal.

I hope this helps. Again, if you need any clarification, please don't
hesistate to request it before rating my answer.


Best wishes!
elmarto