1. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Testing a Claim about a Proportion Section 7-5 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

2. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 Assumptions when Testing a Claim about a Population Proportion or Percentage 1) Binomial experiment conditions are satisfied (Section 4-3)

3. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 Definitions  Binomial Experiment 1.The experiment must have a fixed number of trials. 2.The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.) 3.Each trial must have all outcomes classified into two categories. 4.The probabilities must remain constant for each trial.

4. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 Assumptions when Testing a Claim about a Population Proportion or Percentage 1) Binomial experiment conditions are satisfied (Section 4-3) 2) np  5 and nq  5, so the binomial distribution of sample proportions can be approximated by a normal distribution with µ = np and  = npq

5. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 Notation p = population proportion (given in the null hypothesis) q = 1 – p  n = number of trials p = x/n (sample proportion)

6. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 Test Statistic for Testing a Claim about a Proportion p – p pq n z = 

7. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 x – µ x – np n n p – p Test Statistic for Testing a Claim about a Proportion p – p pq n z =  z = = = = x np  npq pq n n 

8. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 p is sometimes given directly “10% of the observed sports cars are red” is expressed as p = 0.10  

9. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9 (determining the sample proportion of households with cable TV) p is sometimes given directly “10% of the observed sports cars are red” is expressed as p = 0.10 p sometimes must be calculated “96 surveyed households have cable TV and 54 do not” is calculated as p = = = 0.64 x n 96 150    

10. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10 CAUTION  When the calculation of p results in a decimal with many places, store the number on your calculator and use all the decimals when evaluating the z test statistic.  Large errors can result from rounding p too much.  

11. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 All three methods 1) Traditional method 2) P-value method 3) Confidence intervals and the testing procedure Step 1 to Step 8 in Section 7-3 are still valid, except that the test statistic (still the z score by CLT) is calculated using a different formula.

12. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12 Example To test a coin is a fair coin or not, John tossed the coin 150 times and recorded total 68 times that the coin landed on tail. Q: Based on the reported result, can we claim it is a fair coin?

13. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 Example Solution Step 1: p = 0.5 Step 2: p = 0.5 Step 4: Select  = 0.05 (significance level) Step 5: The sample proportion is relevant to this test and its corresponding z-score follows approximately normal distribution (np > 5 and nq > 5, by CLT) Step 3: H 0: p = 0.5 versus H 1: p = 0.5

14. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 Central Limit Theorem: Assume the conjecture is true! z =z = Test Statistic: Critical value = -1.96 * 150 * 0.5 * 0.5 + 150 * 0.5 = 62.9975 75 ( z = 0) 63 ( z = -1.96 ) (Step 6) p – p pq n  Fail to reject H 0 Reject H 0 npq = x - np

15. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 Central Limit Theorem: Assume the conjecture is true! z =z = Test Statistic: Critical value = -1.96 * 150 * 0.5 * 0.5 + 150 * 0.5 = 62.9975 75 ( z = 0) 63 ( z = -1.96 ) (Step 6) p – p pq n  Fail to reject H 0 Reject H 0 npq = x - np Sample data: z = -1.14 x = 68 or

16. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16 Example Conclusion: Based on the reported sample result, there is not enough evidence to rejection of the claim that “the coin is a fair coin”. Step 8:

17. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17 Central Limit Theorem: Assume the conjecture is true! z =z = Test Statistic: 75 68 (Step 6) p – p pq n  Fail to reject H 0 Reject H 0 npq = x - np.45333-.5.5*.5/150 = -1.143 P-value = twice the area to the right of the test statistic z = P-value = 2 * 0.1265 = 0.253

18. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18 Central Limit Theorem: Assume the conjecture is true! z =z = Test Statistic: 75 68 (Step 6) p – p pq n  Fail to reject H 0 Reject H 0 npq = x - np.45333-.5.5*.5/150 = -1.143 P-value = twice the area to the right of the test statistic z = P-value = 0.253 > 0.05 Not enough evidence against the null hypothesis