2. M35 C.I. for proportions 1  Department of ISM, University of Alabama, 1995-2003 Proportions How do “polls” work and what do they tell you?

3. M35 C.I. for proportions 2  Department of ISM, University of Alabama, 1995-2003 Objectives  Create confidence intervals for estimating a true population proportion.  Learn how to use a CI for the “difference of two proportions” to test for independence of two categorical variables.

4. M35 C.I. for proportions 3  Department of ISM, University of Alabama, 1995-2003 X = binary variable.  = proportion in the population having the trait. Population Sample Statistical Inference for Proportions p = proportion in sample having trait. n = sample size. ^ p ^

5. M35 C.I. for proportions 4  Department of ISM, University of Alabama, 1995-2003 X = a count of the number of successes in “n” trials. = the proportion of successes. = XnXn = “batting average” Binomial Distribution involved “counts.” Now change the “count” to “proportion” of successes. p

6. M35 C.I. for proportions 5  Department of ISM, University of Alabama, 1995-2003 For the population of all possible sample proportions:  the standard deviation is  (1-  ) n  p = q and the distribution is approximately Normal. p =  p =   the mean is

7. 0 2 4 6 8 10 12 14 16 18 20 22 X X = count of successes in n trials. X ~ Bino( n = 100,  = 0.10).00.02.04.06.08.10.12.14.16.18.20.22 p  p = p = ?  p = ? ^ ^ p = the proportion of successes in n trials. ^ Example 1. = XnXn

8. M35 C.I. for proportions 7  Department of ISM, University of Alabama, 1995-2003 Sampling Distribution of p ^ p ~ N   p = ,   p = ^ ^  (1 –  ) n ^ [ ] if n  > 5 and n(1–  ) > 5; this a refinement of the n  30 rule. The Central Limit Theorem applies because p is a sample average of n Bernoulli values! ^

9. M35 C.I. for proportions 8  Department of ISM, University of Alabama, 1995-2003 Z  2 p (1 – p) ^^ n ( if n  > 5 and n(1–  ) > 5 ) Margin of Error in using p to estimate  at (1–  )100% confidence: m.o.e. = ^

10. M35 C.I. for proportions 9  Department of ISM, University of Alabama, 1995-2003 if n  > 5 and n(1–  ) > 5. (1–  )100% Confidence Interval for  : m.o.e. + – p ^ Z  2 p (1 – p) ^^ n

11. M35 C.I. for proportions 10  Department of ISM, University of Alabama, 1995-2003

12. M35 C.I. for proportions 11  Department of ISM, University of Alabama, 1995-2003  The governor will spend more on promotion of a new program he wants passed, if fewer than 50% of registered voters support it.  In telephone survey of 200 randomly selected registered voters, 82 say they support the proposed program.  Construct a 95% confidence interval for the true proportion of ALL voters who support the proposed program. Example 2:

13. M35 C.I. for proportions 12  Department of ISM, University of Alabama, 1995-2003 95% confidence interval for p: + – Z  2 p ^ p (1 – p) ^^ n p ^ = sample proportion = 82 / 200 =.41 Example 2.

14. M35 C.I. for proportions 13  Department of ISM, University of Alabama, 1995-2003 What can be concluded from this telephone survey?  The value of concern is 50%. Why?  The CI is.342 to.478. .50 is NOT in this CI; therefore, .50 is not a plausible value.  Less than 50% of the registered voters support the proposed program; therefore, spend more on promotion. Example 2.

15. M35 C.I. for proportions 14  Department of ISM, University of Alabama, 1995-2003

16. Random exit poll results: Sue Ellen: 462 votes of 900. Election night; Birmingham; two candidates for mayor. Can we declare Sue Ellen the winner at the.05 level of significance? Hypothesized value is p =.50; no favorite. m.o.e. = p^ =.5133.5133 *.4867 900 =.03266 1.96  Example 3.

17. M35- C.I. for proportions 16  Department of ISM, University of Alabama, 1995-2002 Construct 95% CI: p ± m.o.e. ^.51333 ±.03266 The 95% CI is.48067 to.54599. “I am 95% confident that the true proportion of votes cast for Sue Ellen in the Birmingham mayoral election falls between.4807 and.5460. Statement in L.O.P: Example 3.

18. M35- C.I. for proportions 17  Department of ISM, University of Alabama, 1995-2002 Decision: Does the “hypothesized value” fall in the CI? Therefore,.50 a plausible value; so the election is “too close to call” at the.05 level of significance.. Example 3.

19. M35- C.I. for proportions 18  Department of ISM, University of Alabama, 1995-2002

20. M35- C.I. for proportions 19  Department of ISM, University of Alabama, 1995-2002 How many votes out of 900 did she need to be declared the winner? Use Excel to see.... Example 3.

21. 95% Confidence Intervals for p: X p-hat m.o.e. lower upper 4780.53111 0.03260 0.49851 0.56371 4790.53222 0.03260 0.49962 0.56482 4800.53333 0.03259 0.50074 0.56593 4810.53444 0.03259 0.50186 0.56703 n = 900 53.33% is enough to be declared the winner, at the.05 level of significance! “What if” values of X

22. M35- C.I. for proportions 21  Department of ISM, University of Alabama, 1995-2002 In a survey about banking services, responses were categorized by age and “opinion of services.” Of the 104 respondents that were 30 years or less, 93 stated that the services were “excellent or good.” Of the 46 that were over 30, 36 stated that the services were “excellent or good.” Is there a dependence between age and “opinion of services”? Is there a dependence between age and “opinion of services”? Example 4.

23. M35- C.I. for proportions 22  Department of ISM, University of Alabama, 1995-2002 9311104 361046 12921150 Age 30 or less Over 30 Service Excellent or Good Acceptable or Poor Total Total Example 4.

24. M35- C.I. for proportions 23  Department of ISM, University of Alabama, 1995-2002 p 1 = P( “Excel or Good” | 30 or less) =.894 93 93104= p 2 = P( “Excel or Good” | over 30) =.783 36 36 46 46= Are these conditional probabilities “far enough apart” to call the true population proportions different? Conditional probabilities:  

25. Margin of Error for p 1 - p 2 : p 1 (1 - p 1 ) n 1 n 1 p 2 (1 - p 2 ) n 2 n 2 + Z  /2 m.o.e.= For 95% confidence: (.783)(.217) 46 46(.894)(.106) 104 104 + 1.96 1.96m.o.e.= = 1.96 (.06786) =.1330  Example 4.

26. M35- C.I. for proportions 25  Department of ISM, University of Alabama, 1995-2002 95% Confidence Interval for the difference of two proportions: p 1 - p 2 + m.o.e. (.894 -.783) +.1330. 111 +.1330 ( -.0220, +.2440 ) Example 4.

27. M35- C.I. for proportions 26  Department of ISM, University of Alabama, 1995-2002 Does “zero” fall inside this confidence interval? Then “zero” is a plausible value for the difference of the two proportions. Therefore, the evidence is not strong enough to say a dependence exists. ( -.0220, +.2440 ) Example 4.

28. M35- C.I. for proportions 27  Department of ISM, University of Alabama, 1995-2002 Conclusion: “Age” and “opinion of service” may be independent, at the 95% confidence level, or at the 5% level of significance. Example 4.

29. M35- C.I. for proportions 28  Department of ISM, University of Alabama, 1995-2002 The two SAMPLE proportions, P( “Excel or Good” | 30 or less) =.894 P( “Excel or Good” | over 30) =.783 are “too close” together to conclude that the corresponding POPULATION proportions are different. Example 4.

30. M35- C.I. for proportions 29  Department of ISM, University of Alabama, 1995-2002 Sample Size for Estimating  Problem: What sample size is needed to have a margin or error less than E at (1–  )100% confidence?  n m.o.e. = z  / 2 < E n >  z  / 2 E 2

31. M35- C.I. for proportions 30  Department of ISM, University of Alabama, 1995-2002 What sample size is needed to estimate the mean “actual mpg” with an m.o.e. of 0.2 mpg with 90% confidence for Honda Accords if the pop. std. dev. is 0.88 mpg? m.o.e. = Z  n  2 0.2 = 1.645 0.88 n n = 1.645 2 0.88 2 0.2 2 = 52.39 Recall

32. M35- C.I. for proportions 31  Department of ISM, University of Alabama, 1995-2002 What if  is unknown?  Use a conservative guess (high).  Use s from a pilot study.  Use a very rough guess of   such as  H – L 4

33. M35- C.I. for proportions 32  Department of ISM, University of Alabama, 1995-2002 Sample Size for Estimating Proportions: What sample size is needed to have a margin or error for estimating p less than “E” at (1–  )100% confidence? m.o.e. = E = Z  2 p (1 – p) ^^ n  n = z  / 2 E 2 p (1 – p) ^^ 2 

34. M35- C.I. for proportions 33  Department of ISM, University of Alabama, 1995-2002 But we don’t know p before we take the sample!  Use a conservative guess (one that results in a larger n.)   =.5 is the most conservative.  Values close to.5 are more conservative than those near 0 or 1.  If you know that the true  should be between.20 and.30, then use.30. ^

35. M35- C.I. for proportions 34  Department of ISM, University of Alabama, 1995-2002 Example 5: What is the smallest sample size necessary to estimate proportion of defective parts to within.02 with 95% confidence if  is known to not exceed 4%?