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Chapter 15

Chapter 15. Tests of Significance: The Basics. Significance Testing. Also called “hypothesis testing” Objective: to test a claim about parameter μ Procedure: State hypotheses H 0 and H a Calculate test statistic Convert test statistic to P-value and interpret

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Chapter 15

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  1. Chapter 15 Tests of Significance: The Basics Basics of Significance Testing

  2. Significance Testing • Also called “hypothesis testing” • Objective: to test a claim about parameter μ • Procedure: • State hypotheses H0 and Ha • Calculate test statistic • Convert test statistic to P-value and interpret • Consider significance level (optional) Basics of Significance Testing

  3. Hypotheses • H0(null hypothesis) claims “no difference” • Ha(alternative hypothesis) contradicts the null • Example: We test whether a population gained weight on average… H0: no average weight gain in populationHa: H0 is wrong (i.e., “weight gain”) • Next  collect data  quantify the extent to which the data provides evidence against H0 9: Basics of Hypothesis Testing

  4. One-Sample Test of Mean • To test a single mean, the null hypothesis isH0: μ = μ0, where μ0 represents the “null value” (null value comes from the research question, not from data!) • The alternative hypothesis can take these forms:Ha: μ > μ0 (one-sided to right) orHa: μ < μ0 (one-side to left) orHa: μ ≠ μ0 (two-sided) • For the weight gain illustrative example:H0: μ = 0 Ha: μ > 0 (one-sided)or Ha: μ ≠ μ0 (two-sided)Note: μ0 = 0 in this example Basics of Significance Testing

  5. Figure: Two possible xbars when H0 true Illustrative Example: Weight Gain • Let X ≡ weight gain • X ~N(μ, σ= 1), the value of μ unknown • Under H0, μ = 0 • Take SRS of n = 10 • σx-bar = 1 / √(10) = 0.316 • Thus, under H0x-bar~N(0, 0.316) Basics of Significance Testing

  6. One-Sample z Statistic Take an SRS of size n from a Normal population. Population σ is known. Test H0: μ= μ0 with: For “weight gain” data, x-bar = 1.02, n = 10, and σ = 1 Basics of Significance Testing

  7. P-value • P-value ≡ the probability the test statistic would take a value as extreme or more extreme than observed test statistic, when H0 is true • Smaller-and-smaller P-values → stronger-and-stronger evidence against H0 • Conventions for interpretation P > .10  evidence against H0 not significant .05 < P ≤ .10  evidence marginally significant .01 < P ≤ .05  evidence against H0 significant P ≤ .01  evidence against H0 very significant Basics of Significance Testing

  8. P-Value Convert z statistics to P-value : • For Ha: μ> μ0P = Pr(Z > zstat) = right-tail beyond zstat • For Ha: μ< μ0 P = Pr(Z < zstat) = left tail beyond zstat • For Ha: μ ¹ μ0 P = 2 × one-tailed P-value Basics of Significance Testing

  9. Illustrative Example • z statistic = 3.23 • One-sided P = P(Z > 3.23) = 1−0.9994 = 0.0006 • Highly significant evidence against H0 Basics of Significance Testing

  10. Significance Level • α≡ threshold for “significance” • We set α • For example, if we choose α = 0.05, we require evidence so strong that it would occur no more than 5% of the time when H0 is true • Decision rule P ≤ α statistically significant evidence P >α nonsignificant evidence • For example, if we set α = 0.01, a P-value of 0.0006 is considered significant Basics of Significance Testing

  11. Summary Basics of Significance Testing

  12. Illustrative Example: Two-sided test Hypotheses:H0: μ= 0 against Ha: μ≠ 0 Test Statistic: P-value: P = 2 × Pr(Z > 3.23) = 2 × 0.0006 = 0.0012 Conclude  highly significant evidence against H0 Basics of Significance Testing

  13. Relation Between Tests and CIs • For two-sided tests, significant results at the α-level  μ0 will fall outside (1–α)100% CI • When α = .05  (1–α)100% = (1–.05)100% = 95% confidence • When α = .01, (1–α)100% = (1–.01)100% = 99% confidence • Recall that we tested H0: μ = 0 and found a two-sided P = 0.0012. Since this is significant at α = .05, we expect “0” to fall outside that 95% confidence interval … continued … Basics of Significance Testing

  14. Relation Between Tests and CIs Recall: xbar = 1.02, n = 10, σ = 1. Therefore, a 95% CI for μ = Since 0 falls outside this 95% CI  the test of H0: μ = 0 is significant at α = .05 Basics of Significance Testing

  15. Example II: Job Satisfaction Does the job satisfaction of assembly workers differ when their work is machine-paced rather than self-paced? A matched pairs study was performed on a sample of workers. Workers’ satisfaction was assessed in each setting. The response variable is the difference in satisfaction scores, self-paced minus machine-paced. The null hypothesis “no average difference” in the population of workers. The alternative hypothesis is “there is an average difference in scores” in the population. H0: m = 0 Ha: m ≠ 0 This is a two-sided test because we are interested in differences in either direction. Basics of Significance Testing

  16. Illustrative Example II • Job satisfaction scores follow a Normal distribution with standard deviation s = 60. • Data from 18 workers gives a sample mean difference score of 17. • Test H0: µ = 0 against Ha: µ≠ 0 with Basics of Significance Testing

  17. Illustrative Example II • Two-sided P-value= Pr(Z < -1.20 or Z > 1.20) = 2 × Pr(Z > 1.20)= (2)(0.1151) = 0.2302 • Conclude: 0.2302 chance we would see results this extreme when H0 is true  evidence against H0 not strong (not significant) Basics of Significance Testing

  18. Example II: Conf Interval Method Studying Job Satisfaction A 90% CI for μ is This 90% CI includes 0. Therefore, it is plausible that the true value of m is 0  H0: µ = 0 cannot be rejected at α= 0.10. Basics of Significance Testing

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