Calculate the sample proportion (p).

Here are the solutions for the problems. 1. Determine a 98% confidence interval for the proportion for PDP in the population. Step 1: Calculate the sample proportion (p). The number of voters interviewed is n = 1000. The number of voters who support PDP is x = 349. p = (x)/(n) = (349)/(1000) = 0.349 Step 2: Find the critical z-value (z_/2). For a 98% confidence interval, = 1 - 0.98 = 0.02. So, /2 = 0.01. The critical z-value corresponding to a cumulative probability of 1 - 0.01 = 0.99 is z_0.01 ≈ 2.326. Step 3: Calculate the standard error (SE). SE = sqrt(p)(1-p)n = sqrt((0.349(1-0.349))/(1000)) = sqrt((0.349 × 0.651)/(1000)) = sqrt((0.227199)/(1000)) = sqrt(0.000227199) ≈ 0.01507 Step 4: Calculate the margin of error (ME). ME = z_/2 × SE = 2.326 × 0.01507 ≈ 0.03505 Step 5: Construct the confidence interval. The confidence interval is p ± ME. 0.349 ± 0.03505 (0.349 - 0.03505, 0.349 + 0.03505) (0.31395, 0.38405) Rounding to three decimal places: (0.314, 0.384) The 98% confidence interval for the proportion of PDP supporters is (0.314, 0.384). 2. Construct a 95% confidence interval for the proportion P of defective items produced by the factory. Step 1: Calculate the sample proportion (p). The number of products sampled is n = 75. The number of defective products is x = 11. p = (x)/(n) = (11)/(75) ≈ 0.146667 Step 2: Find the critical z-value (z_/2). For a 95% confidence interval, = 1 - 0.95 = 0.05. So, /2 = 0.025. The critical z-value corresponding to a cumulative probability of 1 - 0.025 = 0.975 is z_0.025 ≈ 1.96. Step 3: Calculate the standard error (SE). SE = sqrt(p)(1-p)n = sqrt((0.146667(1-0.146667))/(75)) = sqrt((0.146667 × 0.853333)/(75)) = sqrt((0.125111)/(75)) = sqrt(0.00166814) ≈ 0.04084 Step 4: Calculate the margin of error (ME). ME = z_/2 × SE = 1.96 × 0.04084 ≈ 0.08005 Step 5: Construct the confidence interval. The confidence interval is p ± ME. 0.146667 ± 0.08005 (0.146667 - 0.08005, 0.146667 + 0.08005) (0.066617, 0.226717) Rounding to three decimal places: (0.067, 0.227) The 95% confidence interval for the proportion of defective items is (0.067, 0.227). 3. Compute a 96% confidence interval for the proportion of people who use Knorr Maggi cube. Step 1: Calculate the sample proportion (p). The number of people questioned is n = 320. The number of people who answered 'yes' is x = 82. p = (x)/(n) = (82)/(320) = 0.25625 Step 2: Find the critical z-value (z_/2). For a 96% confidence interval, = 1 - 0.96 = 0.04. So, /2 = 0.02. The critical z-value corresponding to a cumulative probability of 1 - 0.02 = 0.98 is z_0.02 ≈ 2.054. Step 3: Calculate the standard error (SE). SE = sqrt(p)(1-p)n = sqrt((0.25625(1-0.25625))/(320)) = sqrt((0.25625 × 0.74375)/(320)) = sqrt((0.1905078125)/(320)) = sqrt(0.000595337) ≈ 0.02440 Step 4: Calculate the margin of error (ME). ME = z_/2 × SE = 2.054 × 0.02440 ≈ 0.05011 Step 5: Construct the confidence interval. The confidence interval is p ± ME. 0.25625 ± 0.05011 (0.25625 - 0.05011, 0.25625 + 0.05011) (0.20614, 0.30636) Rounding to three decimal places: (0.206, 0.306) The 96% confidence interval for the proportion of people who use Knorr Maggi cube is (0.206, 0.306). Drop the next question! 📸