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 Math Help > Statistics > Testing Statistical Hypotheses

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# Testing Statistical Hypotheses

In the previous chapter, we found that by computing a confidence interval, we could obtain a range of likely values for the population parameter we're estimating. Not only that, but we could do a heuristic ``test'' to see if claims were correct by seeing if the confidence interval captured the claimed value. For example, a manufacturer claims that the average lifetime of an electronic component is 32 hours. We could take a sample of electronic components of size n and measure their lifetime. By measuring the sample mean and variance, we can compute a 95% confidence interval. If 32 fell within our interval, we said we would believe the claim of the manufacturer. If it didn't fall within the interval, we wouldn't believe the claim. Hypothesis testing/ is a formal way of testing claims such as these and is closely related to confidence intervals.

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• Heuristic Introduction to Hypothesis Testing

# Heuristic Introduction to Hypothesis Testing

Hypothesis testing in science is a lot like the criminal court system in the United States. How do we decide guilt?

1. Assume innocence until ``proven'' guilty.
2. Evidence is presented at a trial.
3. Proof has to be ``beyond a reasonable doubt.''
A jury's possible decision:
• guilty
• not guilty
Note that a jury cannot declare somebody ``innocent,'' just ``not guilty.'' This is an important point. Do juries ever make mistakes?
1. If a person is really innocent, but the jury decides (s)he's guilty, then they've sent an innocent person to jail.
• Type I error.
2. If a person is really guilty, but the jury finds him/her not guilty, a criminal is walking free on the streets.
• Type II error.
In our criminal court system, a Type I error is considered more important than a Type II error, so we protect against a Type I error to the detriment of a Type II error. This is the same as in statistics.

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• Null and Alternative Hypotheses
• ## Null and Alternative Hypotheses

Science, in general, operates by disproving/ unsatisfactory hypotheses and proposing new-and-improved hypotheses which are testable. The approach we take in statistics is exactly this scientific method. We start with a hypothesis which we assume/ is correct. We call this the null hypothesis/ or , and our goal is to reject in favor of the alternative hypothesis, .

• Type I and Type II Errors
• ## Type I and Type II Errors

The kind of errors we can make are

1. Type I:/ Reject when is really true.
2. Type II:/ Fail to reject when is really false.
It is important to emphasize that we can either reject / or fail to reject/ (in the same sense, a jury can only find someone ``guilty'' or ``not guilty,'' not ``innocent''). Some books will call the latter accepting , but we will try to be careful in using terminology.

In the one and two sample situation, we will always have three forms of :

Note that hypotheses are always about population parameters. The first hypothesis above, , is called a two-sided/ or two-tailed/ test, while the second and third tests are one-sided/ or one-tailed/ hypotheses.

• Review: Hypothesis Testing Facts
• ## Review: Hypothesis Testing Facts

• Hypotheses:
• Null Hypothesis/ : The accepted explanation, status quo. This is what we're trying to disprove.
• Alternate Hypothesis/ : What the researcher or scientist thinks might really be going on, a (possibly) better explanation than the null.
• Test:
• The goal of the test is to reject in favor of . We do this by calculating a test statistic/ and comparing its value with a value from a table in the book, the critical value.
• If our test statistic is more extreme than our critical value, then it falls within the rejection region/ of our test and we reject . We can set up the rejection region before computing our test statistic.
• Decisions:
• Reject .
• Fail to reject .
• Errors:
• Type I:/ Reject when is really true.
• Type II:/ Fail to reject when is really false.

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• General Method for Hypothesis Testing
• # General Method for Hypothesis Testing

We will generally use the following steps in hypothesis testing:

1. Identify from a word problem which category we're in (what the appropriate test statistic is).
2. Determine and .
3. Set up the rejection region by looking up the critical value in the appropriate table.
4. Calculate the test statistic.
5. Draw our conclusion: reject or fail to reject .
6. Interpret our results -- say in words what our conclusion means.
Thus, just like we did using confidence intervals, all we have to do is decide which test to use in which situation.

• Reporting the p-value of a test
• # Reporting the p-value of a test

Often, statisticians will report their test result as a p-value. The p-value indicates the chance that one would obtained a test statistic which is more extreme than the observed one when the is true. The rule is always that we reject if The formula for p-value is given in the next section. See the Tests of Significance concept lab for more about p-values.

• Formulas
• # Formulas

The formulas for the 11 cases considered in the `Calculating Tests of Hypotheses' concept lab are given in the table at the end of this chapter. For some examples, see the chapter for that lab.

• Computer Lab
•
• Concept Lab

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