9 Hypothesis Tests, Lecture notes of Statistics

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Hypothesis Tests

(Ch 9.1 9.3, 9.5 9.9)

Statistical Hypotheses

Statistical hypothesis: a claim about the value of a parameter or population characteristic. Examples:

• H: μ = 75 cents, where μ is the true population average of daily per student candy+soda expenses in US high schools
• H: p < .10, where p is the population proportion of defective helmets for a given manufacturer
• If μ 1 and μ 2 denote the true average breaking strengths of two different types of twine, one

hypothesis might be the assertion that μ 1 – μ 2 = 0, or

another is the statement μ 1 – μ 2 > 5

Components of a Hypothesis Test

1. Formulate the hypothesis to be tested. 2. Determine the appropriate test statistic and calculate it using the sample data. 3. Comparison of test statistic to critical region to draw initial conclusions. 4. Calculation of p value. 5. Conclusion , written in terms of the original problem.

1. Null vs Alternative Hypotheses

In any hypothesis testing problem, there are always two competing hypotheses under consideration:

1. The status quo (null) hypothesis
2. The research (alternative) hypothesis The objective of hypothesis testing is to decide, based on sample information, if the alternative hypotheses is actually supported by the data. We usually do new research to challenge the existing (accepted) beliefs.

1. Null vs Alternative Hypotheses

Why be so committed to the null hypothesis?

• Sometimes we do not want to accept a particular assertion unless (or until) data can show strong support
• Reluctance (cost, time) to change Example: Suppose a company is considering putting a new type of coating on bearings that it produces. The true average wear life with the current coating is

known to be 1000 hours. With μ denoting the true average

life for the new coating, the company would not want to make any (costly) changes unless evidence strongly

suggested that μ exceeds 1000.

1. Null vs Alternative Hypotheses

An appropriate problem formulation would involve testing

H 0 : μ = 1000 against Ha : μ > 1000.

The conclusion that a change is justified is identified with Ha , and it would take conclusive evidence to justify rejecting H 0 and switching to the new coating. Scientific research often involves trying to decide whether a current theory should be replaced, or “elaborated upon.”

Components of a Hypothesis Test

1. Formulate the hypothesis to be tested. 2. Determine the appropriate test statistic and calculate it using the sample data. 3. Comparison of test statistic to critical region to draw initial conclusions. 4. Calculation of p value. 5. Conclusion , written in terms of the original problem.

2. Test Statistics

A test statistic is a rule, based on sample data, for deciding whether to reject H 0. The test statistic is a function of the sample data that will be used to make a decision about whether the null hypothesis should be rejected or not.

2. Test Statistics

Which test statistic is “best”?? There are an infinite number of possible tests that could be devised, so we have to limit this in some way or total statistical madness will ensue! Choice of a particular test procedure must be based on the probability the test will produce incorrect results.

2. Errors in Hypothesis Testing

Definition

• A type I error is when the null hypothesis is rejected, but it is true.
• A type II error is not rejecting H 0 when H 0 is false. This is very similar in spirit to our diagnostic test examples
• False negative test = type I error
• False positive test = type II error

2. Type I errors

Usually: Specify the largest value of α that can be

tolerated, and then find a rejection region with that α.

The resulting value of α is often referred to as the

significance level of the test. Traditional levels of significance are .10, .05, and .01, though the level in any particular problem will depend on the seriousness of a type I error— The more serious the type I error, the smaller the significance level should be.

2. Errors in Hypothesis Testing

We can also obtain a smaller value of α the probability that

the null will be incorrectly rejected – by decreasing the size of the rejection region.

However, this results in a larger value of β for all parameter

values consistent with Ha.

No rejection region that will simultaneously make both α

and all β ’ s small. A region must be chosen to strike a

compromise between α and β.

Components of a Hypothesis Test

1. Formulate the hypothesis to be tested. 2. Determine the appropriate test statistic and calculate it using the sample data. 3. Comparison of test statistic to critical region to draw initial conclusions. 4. Calculation of p value. 5. Conclusion , written in terms of the original problem.

3. Critical region

Rejection regions for z tests: (a) upper tailed test;; (b) lower tailed test;; (c) two tailed test