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Always be mindful of the kindness and not the faults of others.

Always be mindful of the kindness and not the faults of others. Categorical Data. Sections 10.1 to 10.5 Estimation for proportions Tests for proportions Chi-square tests. Example.

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Always be mindful of the kindness and not the faults of others.

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  1. Always be mindful of the kindness and not the faults of others.

  2. Categorical Data Sections 10.1 to 10.5 Estimation for proportions Tests for proportions Chi-square tests

  3. Example • Researchers in the development of new treatments for cancer patients often evaluate the effectiveness of new therapies by reporting the proportion of patients who survive for a specified period of time after completion of the treatment. A new treatment of 870 patients with lung cancer resulted in 330 survived at least 5 years.

  4. Example • Estimate p, the proportion of all patients with lung cancer who would survive at least 5 years after being administered this treatment • How much would you estimate the proportion as?

  5. Distribution of Sample Proportion • Y: the number of successes in the n trials (independent and identical trials) • What’s the distribution of Y? • Sample proportion,

  6. Distribution of Sample Proportion • When np ≥ 5 andn(1-p) ≥ 5, thedistribution of Y can be approximated by a normal distribution. • (approximate) (1-a) Confidence Interval forp: • Optional: (exact) C.I. for p for small sample

  7. Sample Size Where E is the largest tolerable error at (1-a)confidence level.

  8. Test for a Large Sample • When np0≥ 5 andn(1-p0) ≥ 5, the test statistic is:

  9. Inference about 2 Proportions Notation:

  10. Estimation for p1-p2 • Point estimate:

  11. Estimation for p1-p2 • (1-a) Confidence Interval for two large samples:

  12. Example 10.6 • A company markets a new product in the Grand Rapids and Wichita. • In Grand Rapids, the company’s advertising is based entirely on TV commercials. • In Wichita, based on a balanced mix of TV, radio, newspaper, and magazine. • 2 months after the ad campaign begins, the company conducts surveys to determine consumer awareness of the product.

  13. Example 10.6: Data Set Q: Calculate a 95% C.I. for the regional difference in the proportion of all consumers who are aware of the product.

  14. Example 10.6 (conti.) • Conduct a test at a=0.05 to verify if there are >10% more Wichita consumers than Grand Rapids consumers aware of the product.

  15. Test for p1-p2 (2 Large Samples) • When n1p1 ≥ 5 andn1(1-p1) ≥ 5; n2p2 ≥ 5 andn2(1-p2) ≥ 5, the test statistic of Ho: p1-p2=d is • Optional: Fisher Exact Test (p.511)

  16. Minitab • Z test for one proportion: Stat >> Basic Statistics >>1 proportion • Z test for two proportions: Stat >> Basic Statistics >>2 proportion

  17. Chi-Square Goodness of Fit Test • More than two possible outcomes per trial  the multinomial experiment • The experiment consists of n identical trials. • Each trial results in one of k outcomes with probabilities p1, p2,...,pk. • Y=(Y1,…,Yk); Yi = the # of outcome i.

  18. Chi-square Goodness of Fit Test Goal: We are interested in testing a hypothesized distribution of Y (i.e. a set of pi’s values). • Hypotheses: Ho: pi = pio for all i vs. Ha: Ho is false

  19. Chi-square Goodness of Fit Test • Test Statistic: ni = the observed Yi Ei = the expected Yi = npio

  20. Chi-square Goodness of Fit Test • Rejection Region: Reject Ho if where df=k-1. • Note: This test can be trusted only when 80% of more cells of the Ei’s are at least 5.

  21. Example 10.10

  22. Minitab: Stat >> Tables >> Chi-Square Goodness-of-Fit Test(One Variable) Example 10.11

  23. Contingency Table(Example 10.12)

  24. Contingency Table • 2 categorical variables: row and column indexed by i and j, respectively • If they are independent, then

  25. Test for Independence of 2 Var’s • Hypotheses: Ho: the row and column variables are independent Ha: they are dependent • Test Statistic:

  26. Test for Independence of 2 Var’s • Rejection Region: Reject Ho if where df=(r-1)(c-1). • Note: This test can be trusted only when 80% of more cells of the are at least 5.

  27. Minitab: Stat >> Tables >> Chi-Square Test(Two-Way Table in Worksheet) Example 10.12 Chi-Square Test: C1, C2, C3, C4 Expected counts are printed below observed counts Chi-Square contributions are printed below expected counts C1 C2 C3 C4 Total 1 15 32 18 5 70 7.78 26.25 21.39 14.58 6.706 1.260 0.537 6.298 2 8 29 23 18 78 8.67 29.25 23.83 16.25 0.051 0.002 0.029 0.188 3 1 20 25 22 68 7.56 25.50 20.78 14.17 5.688 1.186 0.858 4.331 Total 24 81 66 45 216 Chi-Sq = 27.135, DF = 6, P-Value = 0.000

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