A manufacturer of televisons set claims that 95% of its sets last at
least five years without needing a single repair.  To test this claim
a consumer group selected a sample of 400 consumers who owned the
television set for five years.  Of these 400 consumers, 316 say that
their television sets did not need repair, while 84 say that their
televison set did need at least one repair.

a.  Letting p be the proportion of television sets that last five
years without a single repair, set up the null and alternate
hypothesis tha the consumer group should use to attempt to show the
manaufacturer's claim is false.

b.  Use rejection points and the previously given sample information
to test the hypothesis you set up in part A by setting a equal to .10,
.05, .01, and .001.  How much evidence is there that the
manaufacturer's claim is false?

c.  Do you think the results of the consumer group's survey have
practical importance?  Explain your opinion.

Reviewing, need guidance to make sure on the right track.  Need by 23
Jul 10 PM est.  Thanks

a)p - proportion of television sets that last five years without a single repair

H0: p<= 0.05 (A manufacturer of televisions is right)
H1: p > 0.05 (A manufacturer of televisions is wrong)

b) We need to find a standard deviation of our sample

SIGMAp = Sqrt(Pi*(1-Pi)/n)

n - is a number of recipients of our sample
Pi - is a probability of our hypothesis

SIGMAp = Sqrt(0.05*0.95/400) = 0.011

Let's find the Z-critical of our sample. If this Z-critical value
will be larger than an value in normal distribution (appropriately to
level of rejection)
we reject H0 and accept H1. If not, this means that H0 is a correct hypothesis.

To calculate Z-critical we need to know first what is the proportion (probability)
of television sets that last five years with at least a single repair
in our sample.
n=400
needed at least one repair = 84
p = 84/400 = 0.21

Now let's calculate Z-critical:
p - proportion in sample that we needed to repair
Pi - proportion in our hypothesis

Z-critical = (p-Pi)/SIGMAp
Z-critical = (0.21-0.05)/0.011 = 15.54

Immediately we can see that it's a large value and there are very high
chances we reject H0.
The value is so high that there is almost any doubt that H0 is wrong.

Formally, according to table of normal distribution or excel function
NORMSINV we get:

rejection level 0.1 -> z = 1.28
rejection level 0.05 -> z = 1.64
rejection level 0.01 -> z = 2.33
rejection level 0.001 -> z = 3.09

You can see that the higher the rejection level the greater z value.
This means that
the higher the rejection level the less chances we want to take to
reject the correct H0.
Remember, that we will reject HO only if our Z-critical is greater than table-Z.

Anyway, 15.54 is greater than any possible Z-table value, so we reject H0. 

The button line is that the manufacturer's claim is false for sure. In
medicine experiments, the rejection
level is usually 5%. Here we rejected the H0 for rejection level 0.1%
and we are far bellow Z-critical.
We have a prove that the manufacturer is wrong by all means.

c)I think that if the consumer survey was holds according to all
survey rules (i.e. we may be confident with these results),
they may have a practical importance: I would not make much confident
to this manufacturer since he doesn't make truth.
The results are so clear and firm, that there is no chance that in
next survey you get something, that won't permit you to reject H0.

Thank you for your input, with what you said below, I believe I am on
the right track.

Hello vadim1407-ga

After going over my calculatioins, I have a question.  You stated that
84 were in need of repair.

To calculate Z-critical we need to know first what is the proportion (probability)
of television sets that last five years with at least a single repair
in our sample.
n=400
needed at least one repair = 84
p = 84/400 = 0.21

Now let's calculate Z-critical:
p - proportion in sample that we needed to repair
Pi - proportion in our hypothesis

I am a bit confused because shouldn't the porportion should be the
sets in the last five years without a repair (316)? The questions
asked

a.  Letting p be the proportion of television sets that last five
years without a single repair

Hi !
How you define the p (proportion) changes nothing. It's symetrical.
Anyway, if you want the p to denote the proportion of non-repaired TV,
you get it as follows:
H0: p >=0.95 (A manufacturer of televisions is right)
H1: p < 0.95 (A manufacturer of televisions is wrong)

SIGMAp remains the same since it's the same  0.95 and 0.05.

For z-critical you get:
n=400
TV without repair ned = 316
p = 316/400 = 0.79

Z-critical = (Pi-p)/SIGMAp 
since the hypotesis was defined inverse direction we now have Pi-p instead of p-Pi.

Z-critical = (0.95-0.79)/0.011 = 14.54 (it's 14.54 and not 15.54 - my
mistake; in my previous solution it also was 14.54).

As you can see we got the same answer as before.
Hope this clerifies.

By the way, you may find useful the tutorial from the google sponsored link:
http://www.conceptstew.co.uk/PAGES/s4t5content.html

Yes it does, and thank you for your clarification.  I will also look
at the link you suggested.