Question

Supplementary Problem 8.50 Use the following information to compute the confidence interval for the population proportion. a. n = 715 and x = 329, with 95% confidence b. n = 284 and p = .71, with 90% confidence c. n = 1250 and p = .48, with 95% confidence d. n = 457 and x = 270, with 98% confidence (Round your answers to 4 decimal places.) VI VI VI VI VI VI VI VI

Answer 1

Solution: Part-A: n= 715 x= 329 Sample proportion: X 329 =-= - = 0.46014 Standard Error: p(1-P) Op = 0.46014 x (1 – 0.46014) 715 = 0.01864 For 95% confidence interval, a = 1-0.95 = 0.05 Hence, a/2=0.05/2=0.025 From normal distribution table; 20.025 = 1.96 Margin of error: E = 20.025 X Op E = 1.96 x 0.01864 E = 0.03653 95% confidence interval of population proportion p: = P +E = 0.46014 + 0.03653 Lower limit of 95% confidence interval = 0.46014 – 0.03653 = 0.4236 Upper limit of 95% confidence interval = 0.46014 + 0.03653 = 0.4967 Hence, 95% confidence interval = 0.4236 <p < 0.4967

Part-B: n= 284 P = 0.71 Standard Error: p(1-) Op = 0.71 x (1 -0.71) -= 0.02693 284 Vn For 90% confidence interval, a = 1-0.90 = 0.10 Hence, a/2=0.10/2=0.05 From normal distribution table; 20.05 = 1.645 Margin of error: E = 20.05 X OD E = 1.645 x 0.02693 E = 0.04429 90% confidence interval of population proportion p: = p +E = 0.71 + 0.04429 Lower limit of 90% confidence interval = 0.71 -0.04429 = 0.6657 Upper limit of 90% confidence interval = 0.71 + 0.04429 = 0.7543 Hence, 90% confidence interval = 0.6657 < p < 0.7543

Part-C: n= 1250 P = 0.48 Standard Error: p(1-0) Op = ✓n 0.48 x (1 – 0.48) -= 0.01413 1250 For 95% confidence interval, a = 1-0.95 = 0.05 Hence, a/2=0.05/2=0.025 From normal distribution table; 20.025 = 1.96 Margin of error: E = 20.025 X Op E = 1.96 x 0.01413 E = 0.02770 95% confidence interval of population proportion p: = p +E = 0.48 + 0.02770 Lower limit of 95% confidence interval = 0.48 -0.02770 = 0.4523 Upper limit of 95% confidence interval = 0.48 + 0.02770 = 0.5077 Hence, 95% confidence interval = 0.4523 <p < 0.5077

Part-D: n = 457 x = 270 Sample proportion: x 270 p =-= - = 0.59081 en 457 Standard Error: p(1-P) Op = 0.59081 x (1 – 0.59081) 2 = 0.02300 V n 457 For 98% confidence interval, a = 1-0.98 = 0.02 Hence, a/2=0.02/2=0.01 From normal distribution table; 20.01 = 2.326 Margin of error: E = 20.01 X OD E = 2.326 x 0.02300 E = 0.05351 98% confidence interval of population proportion p: = P +E = 0.59081 + 0.05351 Lower limit of 98% confidence interval = 0.59081 -0.05351 = 0.5373 Upper limit of 98% confidence interval = 0.59081 + 0.05351 = 0.6443 Hence, 98% confidence interval = 0.5373 < p < 0.6443