1. 1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach

2. 2 2 Slide IS 310 – Business Statistics n Inferences About the Difference Between Two Population Proportions Two Population Proportions Chapter 10, Part B Statistical Inferences About Means and Proportions with Two Populations

3. 3 3 Slide IS 310 – Business Statistics Inferences About the Difference Between Two Population Proportions n Interval Estimation of p 1 - p 2 n Hypothesis Tests About p 1 - p 2

4. 4 4 Slide IS 310 – Business Statistics n Expected Value Sampling Distribution of where: n 1 = size of sample taken from population 1 n 2 = size of sample taken from population 2 n 2 = size of sample taken from population 2 n Standard Deviation (Standard Error)

5. 5 5 Slide IS 310 – Business Statistics If the sample sizes are large, the sampling distribution If the sample sizes are large, the sampling distribution of can be approximated by a normal probability of can be approximated by a normal probability distribution. distribution. If the sample sizes are large, the sampling distribution If the sample sizes are large, the sampling distribution of can be approximated by a normal probability of can be approximated by a normal probability distribution. distribution. The sample sizes are sufficiently large if all of these The sample sizes are sufficiently large if all of these conditions are met: conditions are met: The sample sizes are sufficiently large if all of these The sample sizes are sufficiently large if all of these conditions are met: conditions are met: n1p1 > 5n1p1 > 5n1p1 > 5n1p1 > 5 n 1 (1 - p 1 ) > 5 n2p2 > 5n2p2 > 5n2p2 > 5n2p2 > 5 n 2 (1 - p 2 ) > 5 Sampling Distribution of

6. 6 6 Slide IS 310 – Business Statistics Sampling Distribution of p 1 – p 2

7. 7 7 Slide IS 310 – Business Statistics Interval Estimation of p 1 - p 2 n Interval Estimate

8. 8 8 Slide IS 310 – Business Statistics Market Research Associates is Market Research Associates is conducting research to evaluate the effectiveness of a client’s new adver- tising campaign. Before the new campaign began, a telephone survey of 150 households in the test market area showed 60 households “aware” of the client’s product. Interval Estimation of p 1 - p 2 n Example: Market Research Associates The new campaign has been initiated with TV and The new campaign has been initiated with TV and newspaper advertisements running for three weeks.

9. 9 9 Slide IS 310 – Business Statistics A survey conducted immediately A survey conducted immediately after the new campaign showed 120 of 250 households “aware” of the client’s product. Interval Estimation of p 1 - p 2 n Example: Market Research Associates Does the data support the position Does the data support the position that the advertising campaign has provided an increased awareness of the client’s product?

10. 10 Slide IS 310 – Business Statistics Point Estimator of the Difference Between Two Population Proportions = sample proportion of households “aware” of the = sample proportion of households “aware” of the product after the new campaign product after the new campaign = sample proportion of households “aware” of the = sample proportion of households “aware” of the product before the new campaign product before the new campaign p 1 = proportion of the population of households p 1 = proportion of the population of households “aware” of the product after the new campaign “aware” of the product after the new campaign p 2 = proportion of the population of households p 2 = proportion of the population of households “aware” of the product before the new campaign “aware” of the product before the new campaign

11. 11 Slide IS 310 – Business Statistics.08 + 1.96(.0510).08 +.10 Interval Estimation of p 1 - p 2 Hence, the 95% confidence interval for the difference Hence, the 95% confidence interval for the difference in before and after awareness of the product is -.02 to +.18. For  =.05, z.025 = 1.96:

12. 12 Slide IS 310 – Business Statistics Hypothesis Tests about p 1 - p 2 n Hypotheses H 0 : p 1 - p 2 < 0 H a : p 1 - p 2 > 0 Left-tailedRight-tailedTwo-tailed We focus on tests involving no difference between the two population proportions (i.e. p 1 = p 2 )

13. 13 Slide IS 310 – Business Statistics Hypothesis Tests about p 1 - p 2 Standard Error of when p 1 = p 2 = p Standard Error of when p 1 = p 2 = p Pooled Estimator of p when p 1 = p 2 = p Pooled Estimator of p when p 1 = p 2 = p

14. 14 Slide IS 310 – Business Statistics Hypothesis Tests about p 1 - p 2 Test Statistic Test Statistic

15. 15 Slide IS 310 – Business Statistics Can we conclude, using a.05 level Can we conclude, using a.05 level of significance, that the proportion of households aware of the client’s product increased after the new advertising campaign? Hypothesis Tests about p 1 - p 2 n Example: Market Research Associates

16. 16 Slide IS 310 – Business Statistics Hypothesis Tests about p 1 - p 2 1. Develop the hypotheses. p -Value and Critical Value Approaches p -Value and Critical Value Approaches H 0 : p 1 - p 2 < 0 H a : p 1 - p 2 > 0 p 1 = proportion of the population of households p 1 = proportion of the population of households “aware” of the product after the new campaign “aware” of the product after the new campaign p 2 = proportion of the population of households p 2 = proportion of the population of households “aware” of the product before the new campaign “aware” of the product before the new campaign

17. 17 Slide IS 310 – Business Statistics Hypothesis Tests about p 1 - p 2 2. Specify the level of significance.  =.05 3. Compute the value of the test statistic. p -Value and Critical Value Approaches p -Value and Critical Value Approaches

18. 18 Slide IS 310 – Business Statistics Hypothesis Tests about p 1 - p 2 5. Determine whether to reject H 0. We cannot conclude that the proportion of households aware of the client’s product increased after the new campaign. 4. Compute the p –value. For z = 1.56, the p –value =.0594 Because p –value >  =.05, we cannot reject H 0. p –Value Approach p –Value Approach

19. 19 Slide IS 310 – Business Statistics Hypothesis Tests about p 1 - p 2 Critical Value Approach Critical Value Approach 5. Determine whether to reject H 0. Because 1.56 < 1.645, we cannot reject H 0. For  =.05, z.05 = 1.645 4. Determine the critical value and rejection rule. Reject H 0 if z > 1.645 We cannot conclude that the proportion of households aware of the client’s product increased after the new campaign.

20. 20 Slide IS 310 – Business Statistics End of Chapter 10 Part B