Stats: Type of Tests
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This document will explain how to determine if the test is a left tail, right tail, or two-tail test.
The type of test is determined by the Alternative Hypothesis ( H1 )


Left Tailed Test
H1: parameter < value
Notice the inequality points to the left
Decision Rule: Reject H0 if t.s. < c.v.


Right Tailed Test
H1: parameter > value
Notice the inequality points to the right
Decision Rule: Reject H0 if t.s. > c.v.


Two Tailed Test
H1: parameter not equal value
Another way to write not equal is < or >
Notice the inequality points to both sides
Decision Rule: Reject H0 if t.s. < c.v. (left) or t.s. > c.v. (right)
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The decision rule can be summarized as follows:
Reject H0 if the test statistic falls in the critical region
(Reject H0 if the test statistic is more extreme than the critical value)


Confidence Intervals as Tests
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Using the confidence interval to perform a hypothesis test only works with a two-tailed test.
• If the hypothesized value of the parameter lies within the confidence interval with a 1-alpha level of confidence, then the decision at an alpha level of significance is to fail to reject the null hypothesis.
• If the hypothesized value of the parameter lies outside the confidence interval with a 1-alpha level of confidence, then the decision at an alpha level of significance is to reject the null hypothesis.
Sounds simple enough, right? It is.
However, it has a couple of problems.
• It only works with two-tail hypothesis tests.
• It requires that you compute the confidence interval first. This involves taking a z-score or t-score and converting it into an x-score, which is more difficult than standardizing an x-score.
Hypothesis Testing Steps
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Here are the steps to performing hypothesis testing
1. Write the original claim and identify whether it is the null hypothesis or the alternative hypothesis.
2. Write the null and alternative hypothesis. Use the alternative hypothesis to identify the type of test.
3. Write down all information from the problem.
4. Find the critical value using the tables
5. Compute the test statistic




6. Make a decision to reject or fail to reject the null hypothesis. A picture showing the critical value and test statistic may be useful.
6. Write the conclusion.


Testing a Single Mean
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You are testing mu, you are not testing x bar. If you knew the value of mu, then there would be nothing to test.
All hypothesis testing is done under the assumption the null hypothesis is true!
I can't emphasize this enough. The value for all population parameters in the test statistics come from the null hypothesis. This is true not only for means, but all of the testing we're going to be doing.
Population Standard Deviation Known
If the population standard deviation, sigma, is known, then the population mean has a normal distribution, and you will be using the z-score formula for sample means. The test statistic is the standard formula you've seen before.



The critical value is obtained from the normal table, or the bottom line from the t-table.
Population Standard Deviation Unknown
If the population standard deviation, sigma, is unknown, then the population mean has a student's t distribution, and you will be using the t-score formula for sample means. The test statistic is very similar to that for the z-score, except that sigma has been replaced by s and z has been replaced by t.
The critical value is obtained from the t-table. The degrees of freedom for this test is n-1.
If you're performing a t-test where you found the statistics on the calculator (as opposed to being given them in the problem), then use the VARS key to pull up the statistics in the calculation of the test statistic. This will save you data entry and avoid round off errors.
General Pattern
Notice the general pattern of these test statistics is (observed - expected) / standard deviation.




Testing a Single Proportion
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You are testing p, you are not testing p hat. If you knew the value of p, then there would be nothing to test.
All hypothesis testing is done under the assumption the null hypothesis is true!
I can't emphasize this enough. The value for all population parameters in the test statistics come from the null hypothesis. This is true not only for proportions, but all of the testing we're going to be doing.
The population proportion has an approximately normal distribution if np and nq are both at least 5. Remember that we are approximating the binomial using the normal, and that the p we're talking about is the probability of success on a single trial. The test statistic is shown in the box to the right.
The critical value is found from the normal table, or from the bottom row of the t-table.
The steps involved in the hypothesis testing remain the same. The only thing that changes is the formula for calculating the test statistic and perhaps the distribution which is used.
General Pattern
Notice the general pattern of these test statistics is (observed - expected) / standard deviation.