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INFERENTIAL STATISTICS- is not practical to ask every single American how he or she
feels about the fairness of the voting procedures. Instead, we query a relatively small number of
Americans, and draw inferences about the entire country from their responses. The Americans
actually queried constitute our sample of the larger population of all Americans. The
mathematical procedures whereby we convert information about the sample into intelligent
guesses about the population fall under the rubric of inferential statistics 3.

A sample is typically a small subset of the population. In the case of voting attitudes, we would
sample a few thousand Americans, drawn from the hundreds of millions that make up the
country. In choosing a sample, it is therefore crucial that it be representative. It must not
overrepresent one kind of citizen at the expense of others. For example, something would be
wrong with our sample if it happened to be made up entirely of Florida residents. (Recall the
controversy surrounding presidential voting in Florida in 2000.) If the sample held only
Floridians, it could not be used to infer the attitudes of other Americans. The same problem
would arise if the sample were comprised only of Republicans. Inferential statistics are based on
the assumption that sampling is random. 

T-TEST-Student's' t Test (For Independent Samples)

Use this test to compare two small sets of quantitative data when samples are collected
independently of one another. When one randomly takes replicate measurements from a
population he/she is collecting an independent sample. Use of a paired t test, to which some
statistics programs unfortunately default, requires nonrandom sampling (see below).

Criteria

 Only if there is a direct relationship between each specific data point in the first set and
one and only one specific data point in the second set, such as measurements on the same
subject 'before and after,' then the paired t test MAY be appropriate.
 If samples are collected from two different populations or from randomly selected
individuals from the same population at different times, use the test for independent
samples (unpaired).
 Here's a simple check to determine if the paired t test can apply - if one sample can have
a different number of data points from the other, then the paired t test cannot apply.

Examples

T-TEST
'Student's' t Test is one of the most commonly used techniques for testing a hypothesis on the
basis of a difference between sample means. Explained in layman's terms, the t test determines a
probability that two populations are the same with respect to the variable tested.

For example, suppose you collected data on the heights of male basketball and football players,
and compared the sample means using the t test. A probability of 0.4 would mean that there is a
40% liklihood that you cannot distinguish a group of basketball players from a group of football
players by height alone. That's about as far as the t test or any statistical test, for that matter, can
take you. If you calculate a probability of 0.05 or less, then you canreject the null hypothesis
(that is, you can conclude that the two groups of athletes can be distinguished by height.

To the extent that there is a small probability that you are wrong, you haven't proven a
difference, though. There are differences among popular, mathematical, philosophical, legal, and
scientific definitions of proof. I will argue that there is no such thing as scientific proof. Please
see my essay on that subject. Don't make the error of reporting your results as proof (or disproof)
of a hypothesis. No experiment is perfect, and proof in the strictest sense requires perfection.

Make sure you understand the concepts of experimental error and single variable statistics before
you go through this part. Leaves were collected from wax-leaf ligustrum grown in shade and in
full sun. The thickness in micrometers of the palisade layer was recorded for each type of leaf.
Thicknesses of 7 sun leaves were reported as: 150, 100, 210, 300, 200, 210, and 300,
respectively. Thicknesses of 7 shade leaves were reported as 120, 125, 160, 130, 200, 170, and
200, respectively. The mean ± standard deviation for sun leaves was 210 ± 73 micrometers and
for shade leaves it was158 ± 34 micrometers. Note that since all data were rounded to the nearest
micrometer, it is inappropriate to include decimal places in either the mean or standard deviation.

For the t test for independent samples you do not have to have the same number of data points in
each group. We have to assume that the population follows a normal distribution (small samples
have more scatter and follow what is called a t distribution). Corrections can be made for groups
that do not show a normal distribution (skewed samples, for example - note that the word 'skew'
has a specific statistical meaning, so don't use it as a synonym for 'messed up').

The t test can be performed knowing just the means, standard deviation, and number of data
points. Note that the raw data must be used for the t test or any statistical test, for that matter. If
you record only means in your notebook, you lose a great deal of information and usually render
your work invalid. The two sample t test yields a statistic t, in which
X-bar, of course, is the sample mean, and s is the sample standard deviation. Note that the
numerator of the formula is the difference between means. The denominator is a measurement of
experimental error in the two groups combined. The wider the difference between means, the
more confident you are in the data. The more experimental error you have, the less confident you
are in the data. Thus the higher the value of t, the greater the confidence that there is a difference.

To understand how a precise probability value can be attached to that confidence you need to
study the mathematics behind the t distribution in a formal statistics course. The value t is just an
intermediate statistic. Probability tables have been prepared based on the t distribution originally
worked out by W.S. Gossett (see below). To use the table provided, find the critical value that
corrresponds to the number of degrees of freedom you have (degrees of freedom = number of
data points in the two groups combined, minus 2). If t exceeds the tabled value, the means are
significantly different at the probability level that is listed. When using tables report the lowest
probability value for which t exceeds the critical value. Report as 'p < (probability value).'

In the example, the difference between means is 52, A = 14/49, and B = 3242.5. Then t = 1.71
(rounding up). There are (7 + 7 -2) = 12 degrees of freedom, so the critical value for p = 0.05 is
2.18. 1.71 is less than 2.18, so we cannot reject the null hypothesis that the two populations have
the same palisade layer thickness. So now what? If the question is very important to you, you
might collect more data. With a well designed experiment, sufficient data can overcome the
uncertainty contributed by experimental error, and yield a significant difference between
samples, if one exists.

When reporting results of a statistical analysis, always identify what data sets you compared,
what test was used, and for most quantitative data report mean, standard deviation, and the
probability values. Make sure the outcome of the analysis is clearly reported. Some spreadsheet
programs include the t test for i ndependent variables as a built-in option. Even without a
built-in option, is is so easy to set up a spreadsheet to do a paired t test that it may not be worth
the expense and effort to buy and learn a dedicated statistics software program, unless more
complicated statistics are needed.

Z-TEST
The Z-test compares sample and population means to determine if there is a significant
difference.
It requires a simple random sample from a population with a Normal distribution and where
where the mean is known.
Calculation
The z measure is calculated as:

z = (x - m) / SE

where x is the mean sample to be standardized, m (mu) is the population mean

where s is the population standard deviation and n is the sample size.

The z value is then looked up in a z-table. A negative z value means it is below the population
mean (the sign is ignored in the lookup table).
Discussion
The Z-test is typically with standardized tests, checking whether the scores from a particular
sample are within or outside the standard test performance.
The z value indicates the number of standard deviation units of the sample from the population
mean.
Note that the z-test is not the same as the z-score, although they are closely related.
See also
Z-score

Hypothesis Testing

Whenever we have a decision to make about a population characteristic, we make a hypothesis.

        m  > 3    

        m    5.

Suppose that we want to test the hypothesis that m  5.  Then we can think of our opponent
suggesting that m = 5.  We call the opponent's hypothesis the null hypothesis and write:

        H0:    m = 5 

and our hypothesis the alternative hypothesis and write

        H1:    m   5

For the null hypothesis we always use equality, since we are comparing m with a previously
determined mean.

For the alternative hypothesis, we have the choices: < , > , or    .

Procedures in  Hypothesis Testing

When we test a hypothesis we proceed as follows:

2. Choose a level of significance.

3. Determine the sample size.  (Same as confidence intervals)

4. Collect data.

5. Calculate z (or t) score.

6. Utilize the table to determine if the z score falls within the acceptance region.

7. Decide to 

a. Reject the null hypothesis and therefore accept the alternative hypothesis or 

b. Fail to reject the null hypothesis and therefore state that there is not enough
evidence to suggest the truth of the alternative hypothesis.