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Q: Statistics ( Answered 5 out of 5 stars,   0 Comments )
Question  
Subject: Statistics
Category: Science > Math
Asked by: thepilk-ga
List Price: $5.00
Posted: 25 Aug 2002 16:46 PDT
Expires: 24 Sep 2002 16:46 PDT
Question ID: 58428
Having learned about the paired t test you realize that you really
should have used the test for the data on software price comparison. 
The data are repeated below:
Top Ten Business               Computability           PC Connection
Software Packages                Price ($)                Price ($)
  Windows 95 Upgrade               88                       95
  Norton Anti-Virus                59                       70
  McAfee ViruScan                  49                       60
  First Aid 97 Deluxe              54                       58
  Clean Sweep III                  37                       37
  Norton Utilities                 68                       75
  Netscape Navigator               45                       40
  MS Office Pro 97 Upgrade        300                      310
  First Aid 97                     32                       35
  Win Fax Pro                      95                       95

(a) Calculate the differences between the prices for each type of
software pkg.  Just looking at the differences, do you think that one
co. charges more than the other?  Why or why not?
(b) Calculate the average difference and the standard deviation of the
differences.
(c) Set up the hypotheses to test whether the mean difference in price
between the two companies is zero.
(d) Assuming that the dtat are normally distributed, at the 0.05 level
significance, is there a mean difference in the price of software for
the two companies?
Answer  
Subject: Re: Statistics
Answered By: stuartwoozle-ga on 26 Aug 2002 14:01 PDT
Rated:5 out of 5 stars
 
Hello there,

I like statistics - and paired t-tests honestly aren't hard to do. Let
me explain how to do them, using your question as an example...

OK then, so we've got ten different software packages, and the price
that two different companies charge for each of these. We want to find
out whether, on the basis of this sample, one company tends to charge
more for computer software than another company. A paired t-test
sounds sensible then - as each company is quoting for the same set of
software.

(a) OK then, we need to work out the difference between the prices
that the two companies charge for each item of software. I'll work out
the difference in terms of the Computability price minus the PC
Connection price.

SOFTWARE PACKAGES               C ($)       PC ($)     DIFF ($)
Windows 95 Upgrade               88           95          -7
Norton Anti-Virus                59           70         -11
McAfee ViruScan                  49           60         -11
First Aid 97 Deluxe              54           58          -4
Clean Sweep III                  37           37           0
Norton Utilities                 68           75          -7
Netscape Navigator               45           40           5
MS Office Pro 97 Upgrade        300          310         -10
First Aid 97                     32           35          -3
Win Fax Pro                      95           95           0

Well that bit was easy... Next we're asked to just *look* at the
differences, and say whether we think that one company charges more
than the other. Well yes, look at all those minus signs - often with
quite high absolute values as well! In fact, there's only one item of
software (Netscape Navigator) for which PC Connection's price is lower
than that of Computability. On two items there's no difference in
price, but for all the remaining seven PC Connection are more
expensive. They certainly *seem* to consistently charge more, but
whether we can say that this is likely to be the case for *all* the
software they stock is a matter for a statistical test...

(b) OK, so we next need to find the average difference. That's easy -
we just need to add up all the differences and divide by the total
number of software packages (10). The mean difference is therefore
-$4.80. We also need the standard deviation. Now, we've got to be
really careful here, as what we're doing is trying to estimate the
standard deviation of the price difference for *all* the software that
these two companies stock. So on a calculator you have to use the
(n-1) formula to get the right answer, which is $5.37 (to 3
significant figures - 3SF).

(It's outside of the scope of this answer to go into *why* this (n-1)
formula is used, but a Google search on "sample standard deviation"
gave a number of good references (which include full formulae),
including:
http://www.quickmba.com/stats/standard-deviation/ )

(c) OK, next we need to define our hypotheses. First we ought to
consider whether we're wanting a one-tailed test or a two-tailed test.
Well let's think about this for a moment. A one-tailed test would be
helpful if we were only interested in the differences being in a
particular *direction* - for example, if we were asked "Is PC
Connection more expensive than Computability?". In this case though,
we're asked to test whether the mean difference in price between the
companies is zero - i.e. whether the two companies' prices differ.
Because we don't mind *who* charges more, just if *someone* does, we
need a two-tailed test.
OK. So our null hypothesis, H0 = "That the mean difference in software
price between Computability and PC Connection is zero - i.e. that they
do not differ in price."
And our experimental hypothesis, H1 = "That the mean difference in
software price between Computability and PC Connection is not zero -
i.e. that they differ in price."

(d) Last bit - we're almost done! OK, normally for a t-test we'd have
too make sure that the diifferences were normally distributed, but
we're told that we can assume that bit. So now we need to calculate
our value of 't'. Well t = (the mean of the differences) / (the
standard error of the mean). "What's the standard error of the mean?"
I hear you ask. Well that's the standard deviation that we've already
worked out, divided by the square root of the number of items. It
actually represents the standard deviation that we'd get if we took
lots and lots of batches of 10 software samples, and kept working out
the average price difference for each batch. The standard error of the
mean would be the standard deviation of *all* of these averages
considered together.

For a symbolic version of the formula for calculating t, take a look
at:
http://www.dianthus.co.uk/statistics/student.htm#Paired

Anyway, we get a t-value of -2.83 (3SF). Last of all we need to know
whether this t-value lies outside of our critical range for the 0.05
level of significance. In order to do this, we also need to know the
number of degrees of freedom in your data, which is equal to (n-1),
where n = number of data pairs = 10 - 1 = 9.

Consulting a table of significances, I see that the critical range for
a 2-tailed t-test at the 0.05 level of significance lies between
-2.262 and 2.262. These tables are also available online at:
http://www.psychstat.smsu.edu/introbook/tdist.htm

Our value of t lies outside of this critical range - i.e. our result
is likely happen on *less* than 5% of trials purely by chance. We can
therefore conclude that, to the 0.05 level of significance, using a
2-tailed paired t-test, that there *is* a mean difference in the price
of software for the two companies. Cool eh?

If you want to carry out further tests, there are a few online t-test
calculators out there, like this one at:
http://www.physics.csbsju.edu/stats/Paired_t-test_NROW_form.html

I hope my answer helped you, and anyone else reading this, to gain a
more thorough knowledge of what t-tests are and how to carry them out.
If you want to read up more though, most basic statistics books go
into this sort of thing very well.

Best wishes,
stuartwoozle-ga
thepilk-ga rated this answer:5 out of 5 stars
This guy is great.  Not only did he provide complete answers, but he
tried to teach me how to do the work myself, including references.

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