Section 6.7 Suppose Y is the number of successes in n independent Bernoulli trials where the success probability on each trial is p. With a sufficiently large sample size (i.e., np > 5 and n(1 – p) > 5), the Central Limit Theorem tells us that (Y / n) – p —————— =  p(1 – p) / n has an approximatedistribution.N(0,1) Y – np —————  np(1 – p) A 100(1 –  )% confidence interval for p can be derived as follows: (Y / n) – p – z  /2  ——————  z  /2 = 1 –   p(1 – p) / n P P(– Y/n – z  /2  p(1 – p) / n  – p  – Y/n + z  /2  p(1 – p) / n ) = 1 – 

P(Y/n – z  /2  p(1 – p) / n  p  Y/n + z  /2  p(1 – p) / n ) = 1 –  P(Y/n – z  /2  (Y/n)(1 – Y/n) / n  p  Y/n + z  /2  (Y/n)(1 – Y/n) / n ) = 1 –  P(– Y/n – z  /2  p(1 – p) / n  – p  – Y/n + z  /2  p(1 – p) / n ) = 1 – 

1. Do Text Exercise The 98% confidence interval for p is < p < 1497 / 5757  (2.326)  (1497 / 5757)(1 – 1497 / 5757) / We are 98% confident that the proportion of Americans who select jogging as a favorite activity is between and

Suppose Y 1 is the number of successes in n 1 independent Bernoulli trials where the success probability on each trial is p 1. Suppose Y 2 is the number of successes in n 2 independent Bernoulli trials where the success probability on each trial is p 2. Finally, suppose Y 1 and Y 2 are observed independently. Then, Y 1 Y 2 — – — = n 1 n 2 E Y 1 Y 2 — – — = n 1 n 2 Var p 1 – p 2 p 1 (1 – p 1 ) p 2 (1 – p 2 ) ———— + ———— n 1 n 2 With sufficiently large sample sizes (i.e., n 1 p 1 > 5, n 1 (1 – p 1 ) > 5, n 2 p 2 > 5, n 2 (1 – p 2 ) > 5), each of Y 1 / n 1 and Y 2 / n 2 has an approximate normal distribution. Consequently, (from Theorem or Class Exercise 5.3-1) we expect that Y 1 / n 1 – Y 2 /n 2 will have an approximate normal distribution, implying that (Y 1 / n 1 – Y 2 / n 2 ) – (p 1 – p 2 ) —————————————–  p 1 (1 – p 1 ) / n 1 + p 2 (1 – p 2 ) / n 2 has an approximatedistribution.N(0,1)

A 100(1 –  )% confidence interval for p 1 – p 2 can be derived as follows: (Y 1 / n 1 – Y 2 / n 2 ) – (p 1 – p 2 ) – z  /2  —————————————  z  /2 = 1 –   p 1 (1 – p 1 ) / n 1 + p 2 (1 – p 2 ) / n 2 P P (Y 1 /n 1 – Y 2 /n 2 ) – z  /2  (Y 1 /n 1 )(1 – Y 1 /n 1 ) / n 1 + (Y 2 /n 2 )(1 – Y 2 /n 2 ) / n 2  p 1 – p 2  (Y 1 /n 1 – Y 2 /n 2 ) + z  /2  (Y 1 /n 1 )(1 – Y 1 /n 1 ) / n 1 + (Y 2 /n 2 )(1 – Y 2 /n 2 ) / n 2 = 1 –  (Note that one-sided confidence intervals are discussed in the text near the end of Section 6.7)

2. Do Text Exercise (a) (b) The 95% confidence interval for p 1 – p 2 is < p 1 – p 2 <  (1.960)  (0.8203)(1 – ) / (0.6088)(1 – ) / We are 95% confident that the difference in proportion of women who believe clothes are too expensive for Group 1 and Group 2 is between and , with a larger proportion for Group 1. A point estimate of p 1 – p 2 is 1009 / 1230 – 207 / 340 = (0.8203)(0.6088)

Add a worksheet to the Excel file Confidence_Intervals (created previously) which displays the limits of a confidence interval for the proportion and the limits of a confidence interval for the difference between two proportions. 3. (a) (1) Create a worksheet named Proportions in the Excel file named Confidence_Intervals as follows: Change the name of Sheet3 to Proportions. (2) Enter the labels displayed in columns A and B. (3) Color the cells A3, A4, A9, A18, A19, A23, and A29 with a light color such as yellow.

Center the display in cells A3:A6, A9:A12, A18:A21, A23:A26, and A29:32. (5) (6) From the main menu, use the options Insert > Name > Define to assign the name total to cell A3, the name successes to cell A4, the name prop to cell A6, the name one_table to cell A10, the name total_1 to cell A18, the name successes_1 to cell A19, the name prop_1 to cell A21, the name total_2 to cell A23, the name successes_2 to cell A24, the name prop_2 to cell A26, the name two_table to cell A30. (7) Enter the following formulas respectively in cells A5:A6: =IF(OR(ISBLANK(total),ISBLANK(successes)),"-",total-successes) =IF(OR(ISBLANK(total),ISBLANK(successes)),"-",successes/total) (8) Enter the following formulas respectively in cells A10:A12: =IF(ISBLANK(A9),"-",NORMINV(1-0.5*(1-A9),0,1)) =IF(AND(total>0,successes>0,A9>0,A9<1),prop-one_table*SQRT(prop*(1-prop)/total),"-") =IF(AND(total>0,successes>0,A9>0,A9<1),prop+one_table*SQRT(prop*(1-prop)/total),"-")

3.-continued (9) Enter the following formulas respectively in cells A20:A21: =IF(OR(ISBLANK(total_1),ISBLANK(successes_1)),"-",total_1-successes_1) =IF(OR(ISBLANK(total_1),ISBLANK(successes_1)),"-",successes_1/total_1) (10) Enter the following formulas respectively in cells A25:A26: =IF(OR(ISBLANK(total_2),ISBLANK(successes_2)),"-",total_2-successes_2) =IF(OR(ISBLANK(total_2),ISBLANK(successes_2)),"-",successes_2/total_2) (11) Enter the following formulas respectively in cells A30:A32: =IF(ISBLANK(A29),"-",NORMINV(1-0.5*(1-A29),0,1)) =IF(AND(total_1>0,successes_1>0,total_2>0,successes_2>0,$A$29>0,$A$29<1), prop_1-prop_2-two_table*SQRT(prop_1*(1-prop_1)/total_1+prop_2*(1-prop_2)/total_2),"-") =IF(AND(total_1>0,successes_1>0,total_2>0,successes_2>0,$A$29>0,$A$29<1), prop_1-prop_2+two_table*SQRT(prop_1*(1-prop_1)/total_1+prop_2*(1-prop_2)/total_2),"-") (12) Save the file as Confidence_Intervals (in your personal folder on the college network).

(b) (c) Use the Excel file Confidence_Intervals to obtain the confidence interval in Class Exercise #1 of this section (Text Exercise 6.7-6). Use the Excel file Confidence_Intervals to obtain the confidence interval in Class Exercise #2 of this section (Text Exercise ).