Hypothesis Testing
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Definitions
Null Hypothesis ( H0 )
Statement of zero or no change. If the original claim includes equality (<=, =, or >=), it is the null hypothesis. If the original claim does not include equality (<, not equal, >) then the null hypothesis is the complement of the original claim. The null hypothesis always includes the equal sign. The decision is based on the null hypothesis.
Alternative Hypothesis ( H1 or Ha )
Statement which is true if the null hypothesis is false. The type of test (left, right, or two-tail) is based on the alternative hypothesis.
Type I error
Rejecting the null hypothesis when it is true (saying false when true). Usually the more serious error.
Type II error
Failing to reject the null hypothesis when it is false (saying true when false).
alpha
Probability of committing a Type I error.
beta
Probability of committing a Type II error.
Test statistic
Sample statistic used to decide whether to reject or fail to reject the null hypothesis.
Critical region
Set of all values which would cause us to reject H0
Critical value(s)
The value(s) which separate the critical region from the non-critical region. The critical values are determined independently of the sample statistics.
Significance level ( alpha )
The probability of rejecting the null hypothesis when it is true. alpha = 0.05 and alpha = 0.01 are common. If no level of significance is given, use alpha = 0.05. The level of significance is the complement of the level of confidence in estimation.
Decision
A statement based upon the null hypothesis. It is either "reject the null hypothesis" or "fail to reject the null hypothesis". We will never accept the null hypothesis.
Conclusion
A statement which indicates the level of evidence (sufficient or insufficient), at what level of significance, and whether the original claim is rejected (null) or supported (alternative).

Stats: Hypothesis Testing
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Introduction
Be sure to read through the definitions for this section before trying to make sense out of the following.
The first thing to do when given a claim is to write the claim mathematically (if possible), and decide whether the given claim is the null or alternative hypothesis. If the given claim contains equality, or a statement of no change from the given or accepted condition, then it is the null hypothesis, otherwise, if it represents change, it is the alternative hypothesis.
The following example is not a mathematical example, but may help introduce the concept.
Example
"He's dead, Jim," said Dr. McCoy to Captain Kirk.
Mr. Spock, as the science officer, is put in charge of statistically determining the correctness of Bones' statement and deciding the fate of the crew member (to vaporize or try to revive)
His first step is to arrive at the hypothesis to be tested.
Does the statement represent a change in previous condition?
• Yes, there is change, thus it is the alternative hypothesis, H1
• No, there is no change, therefore is the null hypothesis, H0
The correct answer is that there is change. Dead represents a change from the accepted state of alive. The null hypothesis always represents no change. Therefore, the hypotheses are:
• H0 : Patient is alive.
• H1 : Patient is not alive (dead).
States of nature are something that you, as a statistician have no control over. Either it is, or it isn't. This represents the true nature of things.
Possible states of nature (Based on H0)
• Patient is alive (H0 true - H1 false )
• Patient is dead (H0 false - H1 true)
Decisions are something that you have control over. You may make a correct decision or an incorrect decision. It depends on the state of nature as to whether your decision is correct or in error.
Possible decisions (Based on H0 ) / conclusions (Based on claim )
• Reject H0 / "Sufficient evidence to say patient is dead"
• Fail to Reject H0 / "Insufficient evidence to say patient is dead"
There are four possibilities that can occur based on the two possible states of nature and the two decisions which we can make.
Statisticians will never accept the null hypothesis, we will fail to reject. In other words, we'll say that it isn't, or that we don't have enough evidence to say that it isn't, but we'll never say that it is, because someone else might come along with another sample which shows that it isn't and we don't want to be wrong.
Statistically (double) speaking ...
State of Nature
Decision H0 True H0 False
Reject H0 Patient is alive,
Sufficient evidence of death Patient is dead,
Sufficient evidence of death
Fail to reject H0 Patient is alive,
Insufficient evidence of death Patient is dead,
Insufficient evidence of death

In English ...
State of Nature
Decision H0 True H0 False
Reject H0 Vaporize a live person Vaporize a dead person
Fail to reject H0 Try to revive a live person Try to revive a dead person

Were you right ? ...
State of Nature
Decision H0 True H0 False
Reject H0 Type I Error
alpha Correct Assessment
Fail to reject H0 Correct Assessment Type II Error
beta

Which of the two errors is more serious? Type I or Type II ?
Since Type I is the more serious error (usually), that is the one we concentrate on. We usually pick alpha to be very small (0.05, 0.01). Note: alpha is not a Type I error. Alpha is the probability of committing a Type I error. Likewise beta is the probability of committing a Type II error.
Conclusions
Conclusions are sentence answers which include whether there is enough evidence or not (based on the decision), the level of significance, and whether the original claim is supported or rejected.
Conclusions are based on the original claim, which may be the null or alternative hypotheses. The decisions are always based on the null hypothesis
Original Claim

Decision H0
"REJECT" H1
"SUPPORT"
Reject H0
"SUFFICIENT" There is sufficient evidence at the alpha level of significance to reject the claim that (insert original claim here) There is sufficient evidence at the alpha level of significance to support the claim that (insert original claim here)
Fail to reject H0
"INSUFFICIENT" There is insufficient evidence at the alpha level of significance to reject the claim that (insert original claim here) There is insufficient evidence at the alpha level of significance to support the claim that (insert original claim here)

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.

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