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Economics 105: Statistics

Economics 105: Statistics. Any questions ? GH 11 and GH 12 due on Friday. What is a Hypothesis?. A hypothesis is a claim (assumption) about a population parameter:. Example: The mean monthly cell phone bill of this city is μ = $42.

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Economics 105: Statistics

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  1. Economics 105: Statistics Any questions? GH 11 and GH 12 due on Friday

  2. What is a Hypothesis? • A hypothesis is a claim (assumption) about a population parameter: Example: The mean monthly cell phone bill of this city is μ = $42 Example: The proportion of adults in this city with cell phones is π = 0.68

  3. The Null Hypothesis, H0 • States the claim or assertion to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( ) • Is always about a population parameter, not about a sample statistic

  4. The Null Hypothesis, H0 (continued) • Begin with the assumption that the null hypothesis is true • Similar to the notion of innocent until proven guilty • Refers to the status quo • Always contains “=” , “≤” or “”sign • May or may not be rejected

  5. The Alternative Hypothesis, H1 • Is the opposite of the null hypothesis • e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ≠ 3 ) • Challenges the status quo • Never contains “=” , “≤” or “”signs • May or may not be proven find evidence in favor of H1 • Is generally the hypothesis that the researcher is trying to prove to find evidence in favor of

  6. Hypothesis Testing Process Claim:the population mean age is 50. (Null Hypothesis: Population H0: μ = 50 ) Now select a random sample X = likely if μ = 50? Is 20 Suppose the sample If not likely, REJECT mean age is 20: X = 20 Sample Null Hypothesis

  7. Reason for Rejecting H0 Sampling Distribution of X X 20 μ= 50 IfH0 is true ... then we reject the null hypothesis that μ = 50. If it is unlikely that we would get a sample mean of this value ... ... if in fact this were the population mean…

  8. Level of Significance,  • Defines the unlikely values of the sample statistic if the null hypothesis is true • Defines rejection region of the sampling distribution • Is designated by , (level of significance) • Typical values are 0.01, 0.05, or 0.10 • Is selected by the researcher at the beginning • Provides the critical value(s) of the test

  9. Level of Significance and the Rejection Region a Level of significance = Represents critical value a a H0: μ = 3 H1: μ≠ 3 /2 /2 Rejection region is shaded Two-tail test 0 H0: μ≤ 3 H1: μ > 3 a 0 Upper-tail test H0: μ≥ 3 H1: μ < 3 a Lower-tail test 0

  10. Hypothesis Testing

  11. Type I & II Error Relationship • Type I and Type II errors cannot happen at the same time • Type I error can only occur if H0 is true • Type II error can only occur if H0 is false If Type I error probability (  ) , then Type II error probability ( β )

  12. Hypothesis Testing for  Pharmaceutical manufacturer is concerned about impurity concentration in pills, not wanting it to be above 3%. From past production runs, it knows that the impurity concentration in the pills is normally distributed with a standard deviation () of .4%. A random sample of 64 pills was drawn and found to have a mean impurity level of 3.07%. Test the following hypothesis at the 5% level on the test statistic scale. Perform the test on the sample statistic scale. What is the p-value for this test? Power if true pop mean = 3.1%? p-value is the lowest significance level at which you can reject H0.

  13. What are the appropriate H0 & H1? • The Federal Trade Commission wants to prosecute General Mills for not filling its cereal boxes with the advertised weight. • Toyota won’t accept a shipment of tires from its supplier if the tires won’t fit their cars.

  14. What are the appropriate H0 & H1? • A professor would like to know if having a stats lab increases student grades relative to a class without a lab. • Ballbearings-R-Us won’t accept a shipment of ball bearings if more than 5% of the shipment is defective. • A firm that sends out advertising flyers wants to convince potential customers (i.e., firms) that it can increase their sales.

  15. Hypothesis Testing for  Pharmaceutical manufacturer is concerned about impurity concentration in pills, not wanting it to be above 3%. From past production runs, it knows that the impurity concentration in the pills is normally distributed with a standard deviation () of .4%. A random sample of 64 pills was drawn and found to have a mean impurity level of 3.07%. Test the following hypothesis at the 5% level on the test statistic scale. Perform the test on the sample statistic scale. What is the p-value for this test? Power if true pop mean = 3.1%? p-value is the lowest significance level at which you can reject H0.

  16. Hypothesis Testing for  Using t Pharmaceutical manufacturer is concerned about impurity concentration in pills, not wanting it to be different than 3%. A random sample of 16 pills was drawn and found to have a mean impurity level of 3.07% and a standard deviation (s) of .6%. Test the following hypothesis at the 1% level on the test statistic scale. Perform the test on the sample statistic scale. What is the p-value for this test? Calculate the 99% confidence interval.

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