8 Testing a Claim Lesson 8 1 The

8 Testing a Claim Lesson 8. 1 The Idea of a Significance Test Statistics and Probability with Applications, 3 rd Edition Starnes & Tabor Bedford Freeman Worth Publishers

The Idea of a Significance Test Learning Targets After this lesson, you should be able to: ü State appropriate hypotheses for a significance test about a population parameter. ü Interpret a P-value in context. ü Make an appropriate conclusion for a significance test based on a P-value. Statistics and Probability with Applications, 3 rd Edition 2

The Idea of a Significance Test Confidence intervals are one of the two most common types of statistical inference. • Use a confidence interval when your goal is to estimate a population parameter. • The second common type of inference, called a significance test, has a different goal: to test a claim about a parameter. Significance Test A significance test is a formal procedure for using observed data to decide between two competing claims (called hypotheses). The claims are statements about a parameter, like the population proportion p or the population mean µ. Statistics and Probability with Applications, 3 rd Edition 3

Activity I'm a great free-throw shooter! • Statistics and Probability with Applications, 3 rd Edition 4

Activity I'm a great free-throw shooter! 1. Using the pie chart provided by your teacher, label the 80% region “made shot” and the 20% region “missed shot. ” Straighten out one of the ends of a paper clip so that there is a loop on one side and a pointer on the other. On a flat surface, place a pencil through the loop and put the tip of the pencil on the center of the pie chart. Then flick the paper clip and see where the pointed end lands: made shot or missed shot. 2. Flick the paper clip a total of 50 times, and count the number of times that the pointed end lands in the “made shot” region. 3. Compute the sample proportion of made shots in your simulation from Step 2. Plot this value on the class dotplot drawn by your teacher. Statistics and Probability with Applications, 3 rd Edition 5

Activity I'm a great free-throw shooter! • Statistics and Probability with Applications, 3 rd Edition 6

The Idea of a Significance Test A significance test starts with a careful statement of the claims we want to compare. • The claim we seek evidence against is called the null hypothesis, abbreviated H 0. • The claim we hope or suspect to be true instead of the null hypothesis is called the alternative hypothesis. We abbreviate the alternative hypothesis as Ha. Null Hypothesis H 0 , Alternative Hypothesis Ha The claim about the population that we weigh evidence against in a statistical test is called the null hypothesis (H 0). The claim about the population that we are trying to find evidence for is the alternative hypothesis (Ha). Statistics and Probability with Applications, 3 rd Edition 7

The Idea of a Significance Test One-sided alternative hypothesis, Two-sided alternative hypothesis The alternative hypothesis is one-sided if it states that a parameter is greater than the null value or if it states that the parameter is less than the null value. The alternative hypothesis is two-sided if it states that the parameter is different from the null value (it could be either greater than or less than). Statistics and Probability with Applications, 3 rd Edition 8

Do you dance in the rain or just get wet? Stating hypotheses • Statistics and Probability with Applications, 3 rd Edition 9

Do you dance in the rain or just get wet? Stating hypotheses • Statistics and Probability with Applications, 3 rd Edition 10

The Idea of a Significance Test Two cautions about hypotheses: • The hypotheses should express the hope or suspicion we have before we see the data. • Hypotheses always refer to a population, not to a sample. When performing a significance test, we seek evidence against the null hypothesis. To answer the question, “Is the evidence convincing? ” we have to know the likelihood of observing our sample by chance alone. P-value The P-value of a test is the probability of getting evidence for the alternative hypothesis Ha as strong as or stronger than the observed evidence when the null hypothesis H 0 is true. Statistics and Probability with Applications, 3 rd Edition 11

May I have a rain check? Interpreting a P-value • Statistics and Probability with Applications, 3 rd Edition 12

The Idea of a Significance Test The final step in performing a significance test is to draw a conclusion about the competing claims being tested. We make a decision based on the strength of the evidence against the null hypothesis (and in favor of the alternative hypothesis) as measured by the P-value. • Small P-values give convincing evidence for Ha because they say that the observed result is unlikely to occur when H 0 is true. • Large P-values fail to give convincing evidence for Ha because they say that the observed result is likely to occur by chance alone when H 0 is true. How to Make a Conclusion in a Significance Test • If the P-value is small, reject H 0 and conclude that there is convincing evidence for Ha (in context). • If the P-value is large, fail to reject H 0 and conclude that there is not convincing evidence for Ha (in context). Statistics and Probability with Applications, 3 rd Edition 13

May I have the last dance (in the rain)? Making conclusions PROBLEM: Refer to the previous alternate example. What conclusion would you make? The P -value of 0. 0194 is small, so we reject H 0. We have convincing evidence that the true proportion of students at Rob and Justin’s school who have played/danced in the rain is less than 0. 25. Statistics and Probability with Applications, 3 rd Edition 14

LESSON APP 8. 1 Do people kiss the “right” way? 1. State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest. 2. The P-value for the test in Question 1 is 0. 0001. Interpret the Pvalue in context. 3. What conclusion would you make? Statistics and Probability with Applications, 3 rd Edition 15

LESSON APP 8. 1 Do people kiss the “right” way? Statistics and Probability with Applications, 3 rd Edition 16

The Idea of a Significance Test Learning Targets After this lesson, you should be able to: ü State appropriate hypotheses for a significance test about a population parameter. ü Interpret a P-value in context. ü Make an appropriate conclusion for a significance test based on a P-value. Statistics and Probability with Applications, 3 rd Edition 17