Slides Prepared by JOHN S. LOUCKS St. Edward s s University Thomson/South-Western. Slide

Transcription

1 s Preared by JOHN S. LOUCKS St. Edward s s University 1

2 Chater 11 Comarisons Involving Proortions and a Test of Indeendence Inferences About the Difference Between Two Poulation Proortions Hyothesis Test for Proortions of a Multinomial Poulation Test of Indeendence: Contingency Tables

3 Inferences About the Difference Between Two Poulation Proortions Interval Estimation of 1 - Hyothesis Tests About 1-3

4 Samling Distribution of 1 Exected Value E ( ) = 1 1 Standard Deviation (Standard Error) 1 ( 1 1 ) ( 1 ) σ 1 = + n n where: n 1 = size of samle taken from oulation 1 n = size of samle taken from oulation 1 4

5 Samling Distribution of 1 If If the samle sizes are large, the samling distribution of 1 can be aroximated by a normal robability distribution. The samle sizes are sufficiently large if if all of these conditions are met: n 1 1 > 5 n 1 (1-1 ) > 5 n > 5 n (1 - ) > 5 5

6 Samling Distribution of 1 1 ( 1 1 ) ( 1 ) σ 1 = + n n

7 Interval Estimation of 1 - Interval Estimate 1 (1 1 ) (1 ) 1 ± z α / + n n 1 7

8 Interval Estimation of 1 - Examle: Market Research Associates Market Research Associates is conducting research to evaluate the effectiveness of a client s s new adver- tising camaign. Before the new camaign began, a telehone survey of 150 households in the test market area showed 60 households aware of the client s s roduct. The new camaign has been initiated with TV and newsaer advertisements running for three weeks. 8

9 Interval Estimation of 1 - Examle: Market Research Associates A survey conducted immediately after the new camaign showed 10 of 50 households aware of the client s s roduct. Does the data suort the osition that the advertising camaign has rovided an increased awareness of the client s s roduct? 9

10 1 Point Estimator of the Difference Between Two Poulation Proortions 1 = roortion of the oulation of households aware of the roduct after the new camaign = roortion of the oulation of households aware of the roduct before the new camaign = samle roortion of households aware of the roduct after the new camaign = samle roortion of households aware of the roduct before the new camaign = = =

11 Interval Estimation of 1 - For α =.05, z.05 = 1.96:.48(.5).40(.60) ± (.0510) Hence, the 95% confidence interval for the difference in before and after awareness of the roduct is -.0 to

12 Hyothesis Tests about 1 - Hyotheses We focus on tests involving no difference between the two oulation roortions (i.e. 1 = ) H 0 : 1 0 H a: 1 < 0 Left-tailed tailed H 0 : 1 - < 0 H 0 : 1 0 H a: 1 > 0 H a : 1 - > 0 Right-tailed tailed H 0 : 1 = 0 H a: 1 0 Two-tailed 1

13 Hyothesis Tests about 1 - Pooled Estimate of Standard Error of σ 1 = (1 ) + n 1 n where: = n n + n + n

14 Hyothesis Tests about 1 - Test Statistic z = ( ) (1 ) + n n 1 14

15 Examle: Market Research Associates Can we conclude, using a.05 level of significance, that the roortion of households aware of the client s s roduct increased after the new advertising camaign? Hyothesis Tests about 1-15

16 Hyothesis Tests about 1 - -Value and Critical Value Aroaches 1. Develo the hyotheses. H 0 : 1 - < 0 H a : 1 - > 0 1 = roortion of the oulation of households aware of the roduct after the new camaign = roortion of the oulation of households aware of the roduct before the new camaign 16

17 Hyothesis Tests about 1 - -Value and Critical Value Aroaches. Secify the level of significance. α = Comute the value of the test statistic. = 50 (. 48 ) (. 40 ) = = s =. (. )( + ) 50 = (.48.40) 0.08 z = = =

18 Hyothesis Tests about 1 - Value Aroach 4. Comute the value. For z = 1.56, the value = Determine whether to reject H 0. Because value > α =.05, we cannot reject H 0. We cannot conclude that the roortion of households aware of the client s s roduct increased after the new camaign. 18

19 Hyothesis Tests about 1 - Critical Value Aroach 4. Determine the critical value and rejection rule. For α =.05, z.05 = Reject H 0 if z > Determine whether to reject H 0. Because 1.56 < 1.645, we cannot reject H 0. We cannot conclude that the roortion of households aware of the client s s roduct increased after the new camaign. 19

20 Hyothesis (Goodness of Fit) Test for Proortions of a Multinomial Poulation 1. Set u the null and alternative hyotheses.. Select a random samle and record the observed frequency, f i, for each of the k categories. 3. Assuming H 0 is true, comute the exected frequency, e i, in each category by multilying the category robability by the samle size. 0

21 Hyothesis (Goodness of Fit) Test for Proortions of a Multinomial Poulation 4. Comute the value of the test statistic. χ k ( fi ei ) = i = 1 e i where: f i = observed frequency for category i e i = exected frequency for category i k = number of categories Note: The test statistic has a chi-square distribution with k 1 df rovided that the exected frequencies are 5 or more for all categories. 1

22 Hyothesis (Goodness of Fit) Test for Proortions of a Multinomial Poulation 5. Rejection rule: -value aroach: Reject H 0 if -value < α Critical value aroach: Reject H 0 if χ χ α χ χ α where α is the significance level and there are k - 1 degrees of freedom

23 Multinomial Distribution Goodness of Fit Test Examle: Finger Lakes Homes (A) Finger Lakes Homes manufactures four models of refabricated homes, a two-story colonial, a log cabin, a slit-level, level, and an A-frame. A To hel in roduction lanning, management would like to determine if revious customer urchases indicate that there is a reference in the style selected. 3

24 Multinomial Distribution Goodness of Fit Test Examle: Finger Lakes Homes (A) The number of homes sold of each model for 100 sales over the ast two years is shown below. Slit- A- Model Colonial Log Level Frame # Sold

25 Multinomial Distribution Goodness of Fit Test Hyotheses H 0 : C = L = S = A =.5 H a : The oulation roortions are not C =.5, L =.5, S =.5, and A =.5 where: C = oulation roortion that urchase a colonial L = oulation roortion that urchase a log cabin S = oulation roortion that urchase a slit-level level A = oulation roortion that urchase an A-frameA 5

26 Multinomial Distribution Goodness of Fit Test Rejection Rule Reject H 0 if -value <.05 or χ > With α =.05 and k - 1 = 4-1 = 3 degrees of freedom Do Not Reject H 0 Reject H χ 6

27 Multinomial Distribution Goodness of Fit Test Exected Frequencies e 1 =.5(100) = 5 e =.5(100) = 5 e 3 =.5(100) = 5 e 4 =.5(100) = 5 Test Statistic χ ( 30 5) ( 0 5) ( 35 5) ( 15 5) = = =

28 Multinomial Distribution Goodness of Fit Test Conclusion Using the -Value Aroach Area in Uer Tail χ Value (df( = 3) Because χ = 10 is between and , the area in the uer tail of the distribution is between.05 and.01. The -value < α. We can reject the null hyothesis. 8

29 Multinomial Distribution Goodness of Fit Test Conclusion Using the Critical Value Aroach χ = 10 > We reject, at the.05 level of significance, the assumtion that there is no home style reference. 9

30 Test of Indeendence: Contingency Tables 1. Set u the null and alternative hyotheses.. Select a random samle and record the observed frequency, f ij, for each cell of the contingency table. 3. Comute the exected frequency, e ij, for each cell. e ij = (Row i Total)(Column j Total) Samle Size 30

31 Test of Indeendence: Contingency Tables 4. Comute the test statistic. χ = ( f ij e ij ) i j e ij 5. Determine the rejection rule. Reject H 0 if -value < α or χ χ α. where α is the significance level and, with n rows and m columns, there are (n - 1)(m - 1) degrees of freedom. 31

32 Contingency Table (Indeendence) Test Examle: Finger Lakes Homes (B) Each home sold by Finger Lakes Homes can be classified according to rice and to style. Finger Lakes manager would like to determine if the rice of the home and the style of the home are indeendent variables. 3

33 Contingency Table (Indeendence) Test Examle: Finger Lakes Homes (B) The number of homes sold for each model and rice for the ast two years is shown below. For convenience, the rice of the home is listed as either $99,000 or less or more than $99,000. Price Colonial Log Slit-Level A-FrameA < $99, > $99,

34 Contingency Table (Indeendence) Test Hyotheses H 0 : Price of the home is indeendent of the style of the home that is urchased H a : Price of the home is not indeendent of the style of the home that is urchased 34

35 Contingency Table (Indeendence) Test Exected Frequencies Price < $99K > $99K Colonial Log Slit-Level A-Frame A Total Total

36 Contingency Table (Indeendence) Test Rejection Rule With α =.05 and ( - 1)(4-1) = 3 d.f., χ.05 = Reject H 0 if -value <.05 or χ > Test Statistic χ ( ) ( 6 11) ( ) = = =

37 Contingency Table (Indeendence) Test Conclusion Using the -Value Aroach Area in Uer Tail χ Value (df( = 3) Because χ = is between and 9.348, the area in the uer tail of the distribution is between.05 and.05. The -value < α. We can reject the null hyothesis. 37

38 Contingency Table (Indeendence) Test Conclusion Using the Critical Value Aroach χ = > We reject, at the.05 level of significance, the assumtion that the rice of the home is indeendent of the style of home that is urchased. 38

39 End of Chater 11 39