Presentation is loading. Please wait.

Presentation is loading. Please wait.

Two Population Means Hypothesis Testing and Confidence Intervals For Differences in Proportions.

Similar presentations


Presentation on theme: "Two Population Means Hypothesis Testing and Confidence Intervals For Differences in Proportions."— Presentation transcript:

1 Two Population Means Hypothesis Testing and Confidence Intervals For Differences in Proportions

2 SITUATION: 2 Populations Population 1 Population 2 Men - Republican?Women - Republican?TV in Cuba - Good?TV in China - Good?HS >$100K?College > $100K?Trad. Course “A”?Internet Course “A”? Yes = 1 No = 0

3 Notation

4 Individual Responses Proportion of Responses Each Individual Response is: 1 = YES or 0 = NO –Bernoulli Distribution: Mean = p; Variance = pq (where q = (1-p)) The Average or Proportion of n responses (n large) by the Central Limit Theorem is: –Distributed approximately normal with mean = p –Variance = pq/n

5 Differences in Proportions Proportion of “1’s” from Population 1 has: –a normal distribution (approximately) –true mean: p 1 (which we don’t know) –true variance p 1 q 1 /n 1 Proportion of “1’s” from Population 2 has: –a normal distribution (approximately) –true mean: p 2 (which we don’t know) –true variance p 2 q 2 /n 2

6 Distribution of the Difference in Proportions True mean: p 1 - p 2 True variance p 1 q 1 /n 1 + p 2 q 2 /n 2 True standard deviation: SQRT(p 1 q 1 /n 1 + p 2 q 2 /n 2 ) –But since p 1 and p 2 are unknown, what should we use for the standard deviation in confidence intervals and hypothesis tests? For all confidence intervals and for all hypothesis tests except H0: p1- p2 = 0 For hypothesis tests of the form H0: p1- p2 = 0

7 Hypothesis Tests and Confidence Intervals H 0 : p 1 - p 2 = v Where v ≠ 0 H 0 : p 1 - p 2 = 0 Z-Statistics for Difference in Proportions Confidence Interval

8 Example Midas wants to compare customer satisfaction between NY and LA operations. Can it conclude: (1) A greater proportion in LA are satisfied? (2) Customer satisfaction in LA exceeds NY by > 2%? (3) Give a 95% confidence interval for difference in customer satisfaction. Results out of 400 in LA were satisfied 160 out of 200 in NY were satisfied

9 (1) Is a greater proportion in LA? H 0 : p 1 - p 2 = 0 H A : p 1 - p 2 > 0 Select α =.05 Reject H 0 (Accept H A ) if z > z.05 = > 1.645; so it can be concluded that a greater proportion of Midas customers are satisfied customers in LA compared to New York. This is a hypothesis test with v = 0. Use

10 (2) Is Customer Satisfaction more than 2% greater in LA? H 0 : p 1 - p 2 =.02 H A : p 1 - p 2 >.02 Select α =.05. Reject H 0 (Accept H A ) if z > z.05 = > 1.645, so it can be concluded that customer satisfaction in LA exceeds that in New York by more than 2%. This is a hypothesis test with v ≠ 0. Use

11 95% Confidence Interval for the Difference in Proportions

12 Excel – Differences in Proportions =COUNTA(A2:A401) =COUNTA(B2:B201) =F1+F2 =COUNTIF(A2:A401,“YES”) =COUNTIF(B2:B201,“YES”) =F5+F6 =F5/F1 =F6/F2 =F7/F3 =(F9-F10-0)/SQRT(F11*(1-F11)*(1/F1+1/F2)) =1-NORMSDIST(E14) =(F9-F10-.02)/SQRT((F9*(1-F9)/F1)+(F10*(1-F10)/F2)) =1-NORMSDIST(E19) =(F9-F10)-NORMSINV(.975)*SQRT((F9*(1-F9)/F1)+(F10*(1-F10)/F2)) =(F9-F10)+NORMSINV(.975)*SQRT((F9*(1-F9)/F1)+(F10*(1-F10)/F2))

13 Determining Sample Sizes Usual Assumptions: –Sample Sizes equal n 1 = n 2 = n –Take the “worst case scenario” for the standard deviation -- when p 1 = p 2 =.5 (unless you have reason to believe otherwise) –Use the “±” part of the confidence interval How many people need to be surveyed in each city to estimate the difference in customer satisfaction to within ± 4%?

14 No Idea of the Proportions

15 In a Recent Survey 90% in LA and 75% in NY Were Satisfied

16 Review Hypothesis Tests for difference in proportions of the form p 1 - p 2 = 0 –By hand and by Excel Hypothesis Tests for difference in proportions of the form p 1 - p 2 = d –By hand and by Excel Confidence Intervals for difference in proportions –By hand and by Excel Estimating Sample Sizes –No idea of the proportions –Some idea of the proportions


Download ppt "Two Population Means Hypothesis Testing and Confidence Intervals For Differences in Proportions."

Similar presentations


Ads by Google