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Statistical Inference

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

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Statistical Inference

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

  2. Hypothesis • A statement concerning the population usually made in the form of a statement about a population parameter. • Decision making involves choosing between opposing hypotheses. • One is called the Null Hypothesis (H0) and the other is called the Alternative (or Research) Hypothesis (H1)

  3. Null Hypothesis (H0) • A statement about the population asserting the status quo • There is no change, no effect, no difference, etc. • Is usually the opposite of what the researcher is trying to prove • Statement involves =, ≤, or ≥

  4. Alternative Hypothesis (H1) • Research Hypothesis • 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 • Statement involves ≠, <, or >

  5. Overview Statement • H0 is initially assumed to be true. Sample data is collected, and if the sample (statistic) provides sufficient evidence that H0 is false, it is rejected (H1 accepted). Otherwise, we fail to reject H0. • Note that we do not “accept H0 ”. We either “reject H0” or “fail to reject H0”. • i.e. Consider H0 that the world is flat.

  6. 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. H0: Innocent H1: Guilty

  7. Significance Level (a) • 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.

  8. Errors in Hypothesis Testing • Two types of errors in decision making: • Type I (a) - Falsely reject H0 • Type II (b) - Fail to reject H0 when it is false • a and b 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.

  9. Power Analysis • The power (1- b) is the probability of rejecting H0 when it is false, or correctly rejecting. • 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 a and b • Our goals should be control both a and b. For large sample size, a and b can both be small.

  10. Test Statistic & Rejection Region • Test Statistic • A function of the sample data on which the decision to reject or not reject H0 is to be based • Rejection Region • The set of all test statistic values for which H0 will be rejected.

  11. Classical Hypothesis Test • 5 steps in a classical hypothesis test • Hypotheses • Level of Significance (a) • Rejection Region • Test Statistic • Conclusion (Sentence) • Note: If the test statistic is in the rejection region, then H0 is rejected; otherwise H0 is not rejected.

  12. Testing a Population Mean (m) (s known, n≥30) • Test Statistic: • Rejection Region (3 cases of H1): • Two-tailed: For H1: μ ≠ μ0, Reject H0 for |Z| ≥ zα/2 • Left-tailed: For H1: μ < μ0, Reject H0 for Z ≤ -zα • Right-tailed: For H1: μ > μ0, Reject H0 for Z ≥ zα

  13. P-Value • Observed level of significance • Observed type I error rate • Smallest a so that H0 can be rejected • Probability of observing a more extreme (more in favor of H1) value of the test statistic

  14. Hypothesis Testing with the P-Value • 5 steps in the p-value approach to hypothesis testing • Hypotheses • Level of Significance (a) • Test Statistic • P-Value • Conclusion (Sentence) • Note: If the p-value is ≤ a, then H0 is rejected; otherwise H0 is not rejected.

  15. Hypothesis Testing with a Confidence Interval • 5 steps in the p-value approach to hypothesis testing • Hypotheses • Level of Significance (a) • Confidence Interval • A confidence interval with confidence coefficient 1-2a corresponds to a one-sided test with a level of significance. • Conclusion (Sentence) • Note: If the null hypothesis value of the parameter is not in the confidence interval, then H0 is rejected; otherwise H0 is not rejected.

  16. Testing a Population Mean (m) (s unknown) • Test Statistic: • Rejection Region (3 cases of H1) • Two-tailed: For H1: μ ≠ μ0, Reject H0 for |t| ≥ tα/2 • Left-tailed: For H1: μ < μ0, Reject H0 for t ≤ -tα • Right-tailed: For H1: μ > μ0, Reject H0 for t ≥ tα

  17. Testing a Population Proportion (p) • Test Statistic for p: • Rejection Region (3 cases of H1) • Two-tailed: For H1: p ≠ p0, Reject H0 for |Z| ≥ zα/2 • Left-tailed: For H1: p < p0, Reject H0 for Z ≤ -zα • Right-tailed: For H1: p > p0, Reject H0 for Z ≥ zα

  18. Testing a Population Variance (σ2) • Test Statistic for σ2: • Rejection Region (3 cases of H1) • Two-tailed: For H1: σ2 ≠ σ20, Reject H0 for χ2 ≥ χ2α/2 or χ2 ≤ χ21-α/2 • Left-tailed: For H1: σ2 < σ20, Reject H0 for χ2 ≤ χ21-α • Right-tailed: For H1: σ2 > σ20, Reject H0 for χ2 ≥ χ2α

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