Re: Question about the usage of Binomial dist


"Luis A. Afonso" wrote:
mmkhajah@xxxxxxxxx wrote:



*** Hi everyone,


*** I have the following dilemma:

Suppose you have a person A who achieved 65 successes out of 161 trials and another person, B, who has achieved 14 successes out of 32 trials. Which achievement is more valuable, that is, which one is less likely to occur? Of course, one can look at the averages but the average doesn't take
into account the consistency and the number of trials into account.
(A person who wins 500 out of 1000 times has the same average as one who wins 1 time out of 2). I decided to use the Binomial distribution with the MLE for each person. That is,

Person A: p = 65/161
Person B: p = 14/32

Calculating Pr1(K=65) and Pr2(K=14) I got Pr1 < Pr2

But you Pr is the probability of success, so Person A achieved LESS
proportion of successes than Person B. Aren't you drawing the
opposite conclusion below?

so the first person's achievement is more valuable. However, I am not
sure as to whether my inference is correct because the more trials the
person has, the more likely they will win. ***

My response

IF you intend to know what is the most valuable achievement *in terms of conventional Statistics* you can test the Hypotheses that they are REALLY DIFFERENT. Namely

If the NULL HYPOTHESES, H0: they are indistinguishable,

Hypotheses should be written in terms of the population parameters
being tested.
Ho: p1 - p2 = 0.

"must be rejected in favour of the ALTERNATIVE HYPOTHESES, H1, i.e.: H0 cannot be accepted.

There are 3 possible Alternative hypotheses:

Ha: p1 - p2 < 0
Ha: p1 - p2 > 0
Ha: p1 - p2 .NE. 0

Afonso chose the two-tailed test, but that's how that hypothesis should
be written.


The TEST STATISTICS is


______ Z = | pA - pB| / s

pA and pB are the SAMPLE proportions, but the test statistic should be
just Z.

_______s = sqrt ( pA*(1-pA) / 161 + pB*(1-pB) / 34)

This is a common error in TESTING, besides the typo of 34. If one is
constructing a confidence interval for (p1 - p2), then the above is the
correct s to use for the s of the difference of two sample proportions.

But for hypothesis testing, the test statistic MUST be subject to the
condition that Ho is true! And when p1 = p2, the only correct way
to estimate the common variance is to pool the samples into one
estimate of proportion phat = (65+14)/(161+32), then the test stat.
which is of the general form

Z = [ (p1hat - p2hat) - (p1-p2)]/s

This is the general form because one could best (p1-p2) = d, and
p1-p2 would be d, and the corresponding s would be Afonso's s.

will mandate the form (p1hat - p2hat)/s when testing p1-p2 = 0,
with the pooled standard deviation s, and not the incorrect form used
by Afonso, which could be denoted as sd for the standard diviation
of the difference of two proportions, rather than the standard
deviation
of ONE proportion, because when p1=p2, there is only ONE proportion.

This the only place in the elementary course in which the s for a
confidence interval is DIFFERENT from the s for a hypothesis test
of the same parameters!

Many students make the same mistake Afonso did.

which gave 0.3524.

The correct use of the pooled s would have yielded a Z of -.9542,
nearly 3 times as large as the wrong Z, though it's still not large
enough to make a difference in the statistical conclusion.

Afonso's language below (with respect to hypothesis testing) is all
WRONG too. But those are only his secondary errors to the
primary errors of not knowing how to state statistical hypotheses,
and using the wrong test statistic for the test he proposed.

If Afonso had been in my freshman class, and did what he did, I
would have given him 2 points out of 10, for having SOME idea,
about the problem, but had all the details wrong.

-- Reef Fish Bob.




If you had chosen a 95% Confidence Level (which is the most used criterion, though conventional), Z should be AT LEAST 1.960 in order you can claim that H0 must be rejected.
Therefore the conclusion is that pA and pB ARE NOT sufficiently different to which you are allowed to reject H0.
Clearly you show a rather great ignorance in what concerns HYPOTHESES TESTS.

_______licas (Luis A. Afonso)

.

Relevant Pages

  • Re: Question about the usage of Binomial dist
    ... Jack Tomsky wrote: ... exact meaning the parameters are sufficiently ... When he wrote *used the wrong test statistics* ... Afonso claims to be an expert in binomial hypothesis testing. ...
    (sci.stat.math)
  • Re: Question about the usage of Binomial dist
    ... When he wrote *used the wrong test statistics* directing to me it is paranoia in its more pure state: ... It is this part of the BASIC EDUCATION about hypothesis testing, ... you, Luis A. Afonso, is missing. ...
    (sci.stat.math)
  • How to Do It WRONG: Afonso #1 featured.
    ... in "Test of two Independent Proportions: ... The correct approach and actual solution to Afonso #1's problem was ... Two ERRORs in the statement of the TEST STATISTICS: ...
    (sci.stat.math)
  • Re: Question about the usage of Binomial dist
    ... I have taken a statistics course with hypothesis testing; ... When he wrote *used the wrong test statistics* ... Afonso claims to be an expert in binomial hypothesis testing. ... Luis A. Afonso will not be ready for ANYTHING in Lehmann's ...
    (sci.stat.math)
Loading