Confidence and Testing Review

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Review in preparation for exam

Confidence

Without being able, or even desiring, to test
everything, sampling is the greatest tool for statistics. The sample used will never show everything but when properly chosen
and analyzed it will be representative of the entire truth. The real objective of statistics is to be able to make confident
inferences about a population from a sample of that population. The confidence interval can be defined as the sample mean
+/- a margin of error. The smaller that error is the more confidence can be put in the sample.

Hypothesis Testing

Whenever there is a theory about a parameter, it can be tested
via a hypothesis test. The null hypothesis is the presumed result if the alternative hypothesis cannot be proven true. A Type
1 error results when the null is rejected but is in fact true. A Type 2 error is made when the null is not rejected and it
is false.

Hypothesis tests can be done on virtually any population measurement,
but the most common tests are about means and proportions.

Z-tests vs. t-tests

Hypothesis tests for means can use either a z-test or a t-test.
Use a z-test when the population standard deviation is known. Also, use the z-test if the population standard deviation is
not known but the sample is large enough. Without a population standard deviation and only a small sample use the t-test.

Understanding the p-value

A p-value is the probability of making a Type 1 error. If the p-value is smaller than the significance level, than the risk of error is less than the tolerance
for error and so the null hypothesis is rejected. If the p-value is larger than the significance level, the risk is too great
for the tolerance and so do not reject the null hypothesis.

Getting to a Decision

The aim of a study is to prove the alternate. This is true for
both directional and non-directional tests. Since the object is to reject the null and conclude that the alternate is true,
researchers do what they can to help their odds. Remember that a directional test will cut the p-value and help the chances
of rejection, but it must be appropriate to do so. Raising the level of significance gives a wider berth for rejecting the
null, but it is not always practical to tolerate that much error. Finely, increasing the sample size will make a small difference
more significant.

Conclusion

The final exam is part of this module’s
assignment. Review the course carefully, and then take the exam.