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## Hypothesis Testing

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**Hypothesis Testing**Variance known?**Sampling Distribution**• Over-the-counter stock selling prices • calculate average price of all stocks listed [] • take a sample of 25 stocks and record price • calculate average price of the 25 stocks [x-bar] • take all possible samples of size 25 • would all x-bars be equal? • average all the possible x-bars …equals **Sampling Distribution**20 H0 Levine, Prentice-Hall**Sampling Distribution**It is unlikely that we would get a sample mean of this value ... 20 H0 Levine, Prentice-Hall**Sampling Distribution**It is unlikely that we would get a sample mean of this value ... ... if in fact this were the population mean 20 H0 Levine, Prentice-Hall**Sampling Distribution**It is unlikely that we would get a sample mean of this value ... ... therefore, we reject the hypothesis that = 50. ... if in fact this were the population mean 20 H0 Levine, Prentice-Hall**Null Hypothesis**• What is tested • Always has equality sign: , or • Designated H0 • Example ………... H0: 3 Levine, Prentice-Hall**Alternative Hypothesis**• Opposite of null hypothesis • Always has inequality sign: ,, or • Designated H1 • Example • H1: < 3 Levine, Prentice-Hall**Decision**• Reject null hypothesis • Retain, or, fail to reject, null hypothesis • Do not use the term “accept” Levine, Prentice-Hall**p-value**• Probability of obtaining a test statistic more extreme (or than actual sample value given H0 is true • Called observed level of significance • Smallest value of H0 can be rejected • Used to make rejection decision • If p-value , reject H0 Levine, Prentice-Hall**Level of Significance**• Defines unlikely values of sample statistic if null hypothesis is true • Called rejection region of sampling distribution • Designated (alpha) • Typical values are .01, .05, .10 • Selected by researcher at start Levine, Prentice-Hall**Rejection Region (one-tail test)**Sampling Distribution Level of Confidence 1 - Levine, Prentice-Hall**Rejection Region (one-tail test)**Sampling Distribution Level of Confidence 1 - Observed sample statistic Levine, Prentice-Hall**Rejection Region (one-tail test)**Sampling Distribution Level of Confidence 1 - Observed sample statistic Levine, Prentice-Hall**Rejection Regions(two-tailed test)**Sampling Distribution Level of Confidence 1 - Levine, Prentice-Hall**Rejection Regions(two-tailed test)**Sampling Distribution Level of Confidence 1 - Observed sample statistic Levine, Prentice-Hall**Rejection Regions(two-tailed test)**Sampling Distribution Level of Confidence 1 - Observed sample statistic Levine, Prentice-Hall**Rejection Regions(two-tailed test)**Sampling Distribution Level of Confidence 1 - Observed sample statistic Levine, Prentice-Hall**Risk of Errors in Making Decision**• Type I error • Reject true null hypothesis • Has serious consequences • Probability of Type I error is alpha [ ] • Called level of significance • Type II error • Do not reject false null hypothesis • Probability of Type II error is beta [ ] Levine, Prentice-Hall**Decision Results**H0: Innocent Levine, Prentice-Hall**Hypothesis Testing**• State H0 • State H1 • Choose • Choose n • Choose test Levine, Prentice-Hall**Set up critical values**Collect data Compute test statistic Make statistical decision Express decision Hypothesis Testing • State H0 • State H1 • Choose • Choose n • Choose test Levine, Prentice-Hall**Two-tailed z-test**Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes has an average weight = 372.5 grams. The company has specified to be 15 grams. Test at the .05 level. 368 gm. Levine, Prentice-Hall**Two-tailed z-test**Test Statistic: Decision: Conclusion: H0: H1: n Critical Value(s): Levine, Prentice-Hall**Two-tailed z-test**Test Statistic: Decision: Conclusion: H0: = 368 H1: 368 n Critical Value(s): Levine, Prentice-Hall**Two-tailed z-test**Test Statistic: Decision: Conclusion: H0: = 368 H1: 368 .05 n25 Critical Value(s): Levine, Prentice-Hall**Two-tailed z-test**Test Statistic: Decision: Conclusion: H0: = 368 H1: 368 .05 n25 Critical Value(s): Levine, Prentice-Hall**Two-tailed z-test**Test Statistic: Decision: Conclusion: H0: = 368 H1: 368 .05 n25 Critical Value(s): Levine, Prentice-Hall**Two-tailed z-test**Test Statistic: Decision: Conclusion: H0: = 368 H1: 368 .05 n25 Critical Value(s): Do not reject at = .05 Levine, Prentice-Hall**Two-tailed z-test**Test Statistic: Decision: Conclusion: H0: = 368 H1: 368 .05 n25 Critical Value(s): Do not reject at = .05 No evidence average is not 368 Levine, Prentice-Hall**Two-tailed z-test [p-value]]**Z value of sample statistic Levine, Prentice-Hall**Two-tailed z-test [p-value]**p-value is P(z -1.50 or z 1.50) Z value of sample statistic Levine, Prentice-Hall**Two-tailed z-test [p-value]**p-value is P(z -1.50 or z 1.50) Z value of sample statistic Levine, Prentice-Hall**Two-tailed z-test [p-value]**p-value is P(z -1.50 or z 1.50) .4332 From Z table: lookup 1.50 Z value of sample statistic Levine, Prentice-Hall**Two-tailed z-test [p-value]**p-value is P(z -1.50 or z 1.50) .5000- .4332 .0668 .4332 From Z table: lookup 1.50 Z value of sample statistic Levine, Prentice-Hall**Two-tailed z-test [p-value]**p-value is P(z -1.50 or z 1.50) = .1336 .5000- .4332 .0668 .4332 From Z table: lookup 1.50 Z value of sample statistic Levine, Prentice-Hall**Two-tailed z-test [p-value]**1/2 p-value = .0668 1/2 p-value = .0668 1/2 = .025 1/2 = .025 Levine, Prentice-Hall**Two-tailed z-test [p-value]**(p-Value = .1336) ( = .05) Do not reject. 1/2 p-Value = .0668 1/2 p-Value = .0668 1/2 = .025 1/2 = .025 Test statistic is in ‘Do not reject’ region Levine, Prentice-Hall**Two-tailed z-test( known)challenge**You are a Q/C inspector. You want to find out if a new machine is making electrical cords to customer specification: average breaking strength of 70 lb. with = 3.5 lb. You take a sample of 36cords & compute a sample mean of 69.7lb. At the .05level, is there evidence that the machine is not meeting the average breaking strength? Levine, Prentice-Hall**Two-tailed z-test ( known)**Test Statistic: Decision: Conclusion: H0: H1: = n = Critical Value(s): Levine, Prentice-Hall**Two-tailed z-test ( known)**Test Statistic: Decision: Conclusion: H0: = 70 H1: 70 = n = Critical Value(s): Levine, Prentice-Hall**Two-tailed z-test ( known)**Test Statistic: Decision: Conclusion: H0: = 70 H1: 70 = .05 n = 36 Critical Value(s): Levine, Prentice-Hall**Two-tailed z-test ( known)**Test Statistic: Decision: Conclusion: H0: = 70 H1: 70 = .05 n = 36 Critical Value(s): Levine, Prentice-Hall**Two-tailed z-test ( known)**Test Statistic: Decision: Conclusion: H0: = 70 H1: 70 = .05 n = 36 Critical Value(s): Levine, Prentice-Hall**Two-tailed z-test ( known)**Test Statistic: Decision: Conclusion: H0: = 70 H1: 70 = .05 n = 36 Critical Value(s): Do not reject at = .05 Levine, Prentice-Hall**Two-tailed z-test ( known)**Test Statistic: Decision: Conclusion: H0: = 70 H1: 70 = .05 n = 36 Critical Value(s): Do not reject at = .05 No evidence average is not 70 Levine, Prentice-Hall**One-tailed z-test ( known)**• Assumptions • Population is normally distributed • If not normal, can be approximated by normal distribution for large samples Levine, Prentice-Hall**One-tailed z-test ( known)**• Assumptions • Population is normally distributed • If not normal, can be approximated by normal distribution for large samples • Null hypothesis has or sign only Levine, Prentice-Hall**One-tailed z-test ( known)**• Assumptions • Population is normally distributed • If not normal, can be approximated by normal distribution for large samples • Null hypothesis has or sign only • Z-test statistic Levine, Prentice-Hall**One-tailed z-test ( known)**H0:0 H1: < 0 Must be significantly below Levine, Prentice-Hall