Unit 9: Testing a Claim 9.1C—Type I/Type II Errors and the Power of a Test 9.2A—Significance Test for a Population Proportion.

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Unit 9: Testing a Claim 9.1C—Type I/Type II Errors and the Power of a Test 9.2A—Significance Test for a Population Proportion

Objectives 9.1C: Describe and interpret Type I and Type II errors in context Describe how to change the Power of a test

True state of the universe Decision we make based on the sample Ho is True Ho is False Decision we make based on the sample Reject Ho Type I error Correct Decision Fail to reject Ho Type II error

The academy of motion pictures wants to assess the preferences of its customer base. If the majority of Americans prefer to watch their movies at home, then more money will need to be spent on marketing to the home viewer. The relevant hypotheses are: 𝐻 𝑜 :𝑝=0.50 𝐻 𝑎 :𝑝>0.50 Describe a Type I error in the context of this problem. Describe the consequences of a Type II error.

True state of the universe Decision we make based on the sample Ho is True Ho is False Decision we make based on the sample Reject Ho Type I error Correct Decision Fail to reject Ho Type II error P(Type I error) = α P(Type II error) = β But α is also the significance level….and we choose that ourselves!

The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is false (i.e. the probability of not committing a Type II error) Power = 1 − β

How do you get higher power? Increase your sample size Increase α Collect data in a way that reduces variability (stratification, blocking) Move the hypothesized value farther from the true value of the parameter….

Objectives 9.2A: Perform a significance test Write hypotheses Check conditions Perform the mechanics Make a decision linked in context

Conditions: Random Independent Normal (What a shock!)

Example: Is it better to go last? 600 fans of an American Idol type show were shown the audition tapes of 12 people in random order. 59 choose the last tape as their favorite. Does this sample provide support for the claim that it is advantageous to go last?

Are young people at risk for heart disease? Example: Are young people at risk for heart disease? In 2005, researchers studies the physical fitness of adolescents aged 12 to 19 years. Of the 2205 tests, 750 showed a poor level of cardiovascular fitness. Does this sample provide support for the claim that more than 30% of adolescents have a low level of cardiovascular fitness?

Example: Fewer Parolees Land Back in Prison Quote: “Just over 38% of all felons who were released from prison in 2003 landed back in jail by the end of the following year.” Explain why it would not be necessary to carry out a hypothesis test to see if the proportion of felons released in 2003 was less than 40%.