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Introduction to Inference

Introduction to Inference. Tests of Significance Chapter 9. Proof. 925 950 975 1000. Proof. 925 950 975 1000. Definitions. A test of significance is a method for using sample data to make a decision about a population characteristic.

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Introduction to Inference

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  1. Introduction to Inference Tests of Significance Chapter 9

  2. Proof 925 950 975 1000

  3. Proof 925 950 975 1000

  4. Definitions • A test of significance is a method for using sample data to make a decision about a population characteristic. • The null hypothesis, written H0,is the starting value for the decision (i.e. H0 : m = 1000). • The alternative hypothesis, written Ha,states what belief/claim we are trying to determine if statistically significant (Ha : m < 1000).

  5. Examples • Chrysler Concord • H0: m = 8 • Ha: m > 8 • K-mart • H0: m = 1000 • Ha: m < 1000

  6. Chrysler 8

  7. K-mart 1000

  8. Phrasing our decision • In justice system, what is our null and alternative hypothesis? • H0: defendant is innocent • Ha: defendant is guilty • What does the jury state if the defendant wins? • Not guilty • Why?

  9. Phrasing our decision • H0: defendant is innocent • Ha: defendant is guilty • If we have the evidence: • We reject the belief the defendant is innocent because we have the evidence to believe the defendant is guilty. • If we don’t have the evidence: • We fail to reject the belief the defendant is innocent because we do not have the evidence to believe the defendant is guilty.

  10. Chrysler Concord • H0: m = 8 • Ha: m > 8 • p-value = .0134 • We reject H0 since the probability is so small there is enough evidence to believe the mean Concord time is greater than 8 seconds.

  11. K-mart light bulb • H0: m = 1000 • Ha: m < 1000 • p-value = .1078 • We fail to reject H0 since the probability is not very small there is not enough evidence to believe the mean lifetime is less than 1000 hours.

  12. Remember:Inference procedure overview • State the procedure • Define any variables • Establish the conditions (assumptions) • Use the appropriate formula • Draw conclusions

  13. Test of Significance Example • A package delivery service claims it takes an average of 24 hours to send a package from New York to San Francisco. An independent consumer agency is doing a study to test the truth of the claim. Several complaints have led the agency to suspect that the delivery time is longer than 24 hours. Assume that the delivery times are normally distributed with standard deviation (assume s for now) of 2 hours. A random sample of 25 packages has been taken.

  14. Example 1 1-sample z-test m = true mean delivery time Ho: m = 24 Ha: m > 24 Given random sample Given normal distribution At least 250 packages in pop.

  15. 24.85 Example 1 (look, don’t copy) 22.8 23.2 23.6 24 24.4 24.8 25.2

  16. Example 1 1-sample z-test m = true mean delivery time Ho: m = 24 Ha: m > 24 Given random sample Given normal distribution At least 250 packages in pop. let a = .05

  17. Example 1 1-sample z-test m = true mean delivery time Ho: m = 24 Ha: m > 24 Given random sample Given normal distribution At least 250 packages in pop. We reject Ho. Since p-value<a there is enough evidence to believe the delivery time is longer than 24 hours.

  18. Example 1 Another way of thinking about it… If the true mean delivery time is 24 hours, then there is only a 1.66% chance of getting a sample mean of at least 24.85 hours (not very likely to occur).

  19. Wording of conclusion revisit • If I believe the statistic is just too extreme and unusual (P-value < a), I will reject the null hypothesis. • If I believe the statistic is just normal chance variation (P-value > a), I will fail to reject the null hypothesis. reject fail to reject p-value<a, there is p-value>a, there is not We Ho, since the enough evidence to believe…(Ha in context…)

  20. Familiar transition • What happened on day 2 of confidence intervals involving mean and standard deviation? • Switch from using z-scores to using the t-distribution. • What changes occur in the write up?

  21. Example 3 1-sample t-test m = true mean distance Ho: m = 340 Ha: m > 340 Given random sample Given normally distributed. At least 100 missiles in pop. p-value=

  22. t-chart

  23. Example 3 1-sample t-test m = true mean distance Ho: m = 340 Ha: m > 340 Given random sample Given normally distributed. At least 100 missiles in pop. .10<p-value<.15

  24. Example 3 1-sample t-test m = true mean distance Ho: m = 340 Ha: m > 340 Given random sample Given normally distributed. At least 100 missiles in pop. p-value = .1188 We fail to reject Ho. Since p-value>a there is not enough evidence to believe the mean distance traveled is more than 340 miles.

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