Statistical Inference Decision Making (Hypothesis Testing) Decision Making (Hypothesis Testing) A formal method for decision making in the presence of uncertainty. A formal method for decision making in the presence of uncertainty. Does not rely on intuitionDoes not rely on intuition Hypothesis testing answers a specific question about the parameter of interest Hypothesis testing answers a specific question about the parameter of interest Is the mean time for service less than 30 minutes?Is the mean time for service less than 30 minutes? Do a majority of voters support the candidate?Do a majority of voters support the candidate?

Hypothesis A statement concerning the population usually made in the form of a statement about a population parameter. A statement concerning the population usually made in the form of a statement about a population parameter. Decision making involves choosing between opposing hypotheses. Decision making involves choosing between opposing hypotheses. One is called the Null Hypothesis (H 0 ) and the other is called the Alternative (or Research) Hypothesis (H 1 ) One is called the Null Hypothesis (H 0 ) and the other is called the Alternative (or Research) Hypothesis (H 1 )

Null Hypothesis (H 0 ) A statement about the population asserting the status quo A statement about the population asserting the status quo There is no change, no effect, no difference, etc. There is no change, no effect, no difference, etc. Is usually the opposite of what the researcher is trying to prove Is usually the opposite of what the researcher is trying to prove Statement involves =, ≤, or ≥ Statement involves =, ≤, or ≥

Alternative Hypothesis (H 1 ) Research Hypothesis Research Hypothesis A statement about the population asserting change A statement about the population asserting change A statement of what the researcher is trying to prove, or what is believed to be true instead of the null hypothesis A statement of what the researcher is trying to prove, or what is believed to be true instead of the null hypothesis Statement involves ≠, Statement involves ≠,

Overview Statement H 0 is initially assumed to be true. Sample data is collected, and if the sample (statistic) provides sufficient evidence that H 0 is false, it is rejected (H 1 accepted). Otherwise, we fail to reject H 0. H 0 is initially assumed to be true. Sample data is collected, and if the sample (statistic) provides sufficient evidence that H 0 is false, it is rejected (H 1 accepted). Otherwise, we fail to reject H 0. Note that we do not “accept H 0 ”. We either “reject H 0 ” or “fail to reject H 0 ”. Note that we do not “accept H 0 ”. We either “reject H 0 ” or “fail to reject H 0 ”. i.e. Consider H 0 that the world is flat.i.e. Consider H 0 that the world is flat.

Judicial System The decision making process is analogous to the judicial system in America. Innocence is assumed unless there is sufficient evidence (beyond a shadow of doubt) to prove guilt. The decision making process is analogous to the judicial system in America. Innocence is assumed unless there is sufficient evidence (beyond a shadow of doubt) to prove guilt. H 0 : Innocent H 1 : Guilty

Significance Level (  ) The level of significance of a test is the probability of falsely rejecting the null hypothesis. The level of significance of a test is the probability of falsely rejecting the null hypothesis. The decision making criterion is based on controlling this error rate... keeping it sufficiently small. The decision making criterion is based on controlling this error rate... keeping it sufficiently small.

Errors in Hypothesis Testing Two types of errors in decision making: Two types of errors in decision making: Type I (  ) - Falsely reject H 0 Type I (  ) - Falsely reject H 0 Type II (  ) - Fail to reject H 0 when it is false Type II (  ) - Fail to reject H 0 when it is false  and  are inversely proportional, meaning that decreasing one will increase the other.  and  are inversely proportional, meaning that decreasing one will increase the other. Typically, the Type I error rate is set to a moderate level, resulting in a reasonable Type II error rate. Typically, the Type I error rate is set to a moderate level, resulting in a reasonable Type II error rate.

Power Analysis The power (1-  ) is the probability of rejecting H 0 when it is false, or correctly rejecting. The power (1-  ) is the probability of rejecting H 0 when it is false, or correctly rejecting. It measures the ability to prove the research hypothesis when it is in fact true.It measures the ability to prove the research hypothesis when it is in fact true. A power analysis can be used to determine the required sample size for specified  and  A power analysis can be used to determine the required sample size for specified  and  Our goals should be control both  and  For large sample size,  and  can both be small.Our goals should be control both  and  For large sample size,  and  can both be small.

Test Statistic & Rejection Region Test Statistic Test Statistic A function of the sample data on which the decision to reject or not reject H 0 is to be based A function of the sample data on which the decision to reject or not reject H 0 is to be based Rejection Region Rejection Region The set of all test statistic values for which H 0 will be rejected. The set of all test statistic values for which H 0 will be rejected.

Classical Hypothesis Test 5 steps in a classical hypothesis test 5 steps in a classical hypothesis test 1. Hypotheses 2. Level of Significance (  ) 3. Rejection Region 4. Test Statistic 5. Conclusion (Sentence) Note: If the test statistic is in the rejection region, then H 0 is rejected; otherwise H 0 is not rejected.Note: If the test statistic is in the rejection region, then H 0 is rejected; otherwise H 0 is not rejected.

Testing a Population Mean (   known, n≥30  Test Statistic: Test Statistic: Rejection Region (3 cases of H 1 ): Rejection Region (3 cases of H 1 ): 1. Two-tailed:For H 1 : μ ≠ μ 0, Reject H 0 for |Z| ≥ z α/2 2. Left-tailed:For H 1 : μ < μ 0, Reject H 0 for Z ≤ -z α 3. Right-tailed:For H 1 : μ > μ 0, Reject H 0 for Z ≥ z α

P-Value Observed level of significance Observed level of significance Observed type I error rate Observed type I error rate Smallest  so that H 0 can be rejected Smallest  so that H 0 can be rejected Probability of observing a more extreme (more in favor of H 1 ) value of the test statistic Probability of observing a more extreme (more in favor of H 1 ) value of the test statistic

Hypothesis Testing with the P-Value 5 steps in the p-value approach to hypothesis testing 5 steps in the p-value approach to hypothesis testing 1. Hypotheses 2. Level of Significance (  ) 3. Test Statistic 4. P-Value 5. Conclusion (Sentence) Note: If the p-value is ≤ , then H 0 is rejected; otherwise H 0 is not rejected.Note: If the p-value is ≤ , then H 0 is rejected; otherwise H 0 is not rejected.

Hypothesis Testing with a Confidence Interval 5 steps in the p-value approach to hypothesis testing 5 steps in the p-value approach to hypothesis testing 1. Hypotheses 2. Level of Significance (  ) 3. Confidence Interval A confidence interval with confidence coefficient 1-2  corresponds to a one-sided test with  level of significance.A confidence interval with confidence coefficient 1-2  corresponds to a one-sided test with  level of significance. 4. Conclusion (Sentence) Note: If the null hypothesis value of the parameter is not in the confidence interval, then H 0 is rejected; otherwise H 0 is not rejected.Note: If the null hypothesis value of the parameter is not in the confidence interval, then H 0 is rejected; otherwise H 0 is not rejected.

Testing a Population Mean (   unknown  Test Statistic: Test Statistic: Rejection Region (3 cases of H 1 ) Rejection Region (3 cases of H 1 ) 1. Two-tailed:For H 1 : μ ≠ μ 0, Reject H 0 for |t| ≥ t α/2 2. Left-tailed:For H 1 : μ < μ 0, Reject H 0 for t ≤ -t α 3. Right-tailed:For H 1 : μ > μ 0, Reject H 0 for t ≥ t α

Testing a Population Proportion (p) Test Statistic for p: Test Statistic for p: Rejection Region (3 cases of H 1 ) Rejection Region (3 cases of H 1 ) 1. Two-tailed:For H 1 : p ≠ p 0, Reject H 0 for |Z| ≥ z α/2 2. Left-tailed:For H 1 : p < p 0, Reject H 0 for Z ≤ -z α 3. Right-tailed:For H 1 : p > p 0, Reject H 0 for Z ≥ z α

Testing a Population Variance (σ 2 ) Test Statistic for σ 2 : Test Statistic for σ 2 : Rejection Region (3 cases of H 1 ) Rejection Region (3 cases of H 1 ) 1. Two-tailed: For H 1 : σ 2 ≠ σ 2 0, Reject H 0 for χ 2 ≥ χ 2 α/2 or χ 2 ≤ χ 2 1-α/2 1. Left-tailed: For H 1 : σ 2 < σ 2 0, Reject H 0 for χ 2 ≤ χ 2 1-α 2. Right-tailed: For H 1 : σ 2 > σ 2 0, Reject H 0 for χ 2 ≥ χ 2 α