Chapter 9 Testing A Claim 9.1 SIGNIFICANT TESTS: THE BASICS OUTCOME: I WILL STATE THE NULL AND ALTERNATIVE HYPOTHESES FOR A SIGNIFICANCE TEST ABOUT A POPULATION PARAMETER, INTERPRET A P-VALUE IN CONTEXT, DETERMINE WHETHER THE RESULTS OF A STUDY ARE STATISTICAL SIGNIFICANT AND MAKE AN APPROPRIATE CONCLUSION USING A SIGNIFICANCE LEVEL, AND INTERPRET A TYPE I AND TYPE II ERROR IN CONTEXT AND GIVE A CONSEQUENCE OF EACH.

Activity  A basketball player claims to make 80% of the free throws that he attempts. We think he might be exaggerating. To test this claim, we’ll ask him to shoot some free throws – virtually.  I’m a Great Free-Throw Shooter I’m a Great Free-Throw Shooter  What did we learn?  Increasing the sample size helps us be more certain that the difference between the observed proportion of successes and the hypothesized proportion of successes is not due to sampling variability.

What is a significance test?  A significance test is a formal procedure for using observed data to decide between two competing claims (also called hypotheses)  The claims are often statements about a parameter.

Stating Hypotheses

Hypothesis Tests Answer The Question “Do I think this could have happened by chance?”

Stating Hypotheses  Our alternative hypothesis is one-sided because we are interested only in whether the player is overstating his free-throw shooting ability.

Then…What’s a Two-Sided Hypothesis?  At the Hawaii Pineapple Company, managers are interested in the size of the pineapples grown in the company’s fields. Last year, the mean weight of pineapples harvested from one large field was 31 ounces. A different irrigation system was installed in this field after the growing season. Managers wonder how this change will affect the mean weight of pineapples grown in the field this year.  State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.

Let’s Sum That Up  The alternative hypothesis is one-sided if it states that a parameter is larger than or smaller than the null value.  It is two-sided if it states that the parameter is different from the null hypothesis value (it could be either larger or smaller).

Let’s be clear…

Interpreting P -values The P-value measures surprise

P -value

Example

Statistical Significance

BE CAREFUL

Quickly back to our golf example.

Let’s summarize that up a bit…

However!

Recall in Chapter 4

So again… Note: In practice, the most commonly used significance level is α = 0.05

Example

Chapter 9 Testing A Claim 9.1 SIGNIFICANT TESTS: THE BASICS OUTCOME: I WILL STATE THE NULL AND ALTERNATIVE HYPOTHESES FOR A SIGNIFICANCE TEST ABOUT A POPULATION PARAMETER, INTERPRET A P-VALUE IN CONTEXT, DETERMINE WHETHER THE RESULTS OF A STUDY ARE STATISTICAL SIGNIFICANT AND MAKE AN APPROPRIATE CONCLUSION USING A SIGNIFICANCE LEVEL, AND INTERPRET A TYPE I AND TYPE II ERROR IN CONTEXT AND GIVE A CONSEQUENCE OF EACH. Chapters 7-9 Test 3/11 (A) or 3/14 (B)

Hypothesis tests do not always lead to a correct decision.

Type I and Type II Errors  When we draw a conclusion from a significance test, we hope our conclusion will be correct (obviously). But sometimes, it will be wrong.  There are 2 types of mistakes we can make: Type I and Type II errors.

Type I Error  When we reject the null hypothesis when it is actually true  Like if we find someone guilty who is actually innocent

Type II Error  When we fail to reject the null hypothesis when the alternative hypothesis is actually true.  Like when we fail to convict a guilty person

 On the AP exam, it is easy to reverse Type I and Type II errors  Think about it like this, a Type II error happens when you fail “II” reject the null hypothesis when you really should have.

Example

 Suppose that the manager decided to carry out this test using a random sample of 250 orders and a significance level of α = What is the probability of making a Type I error?