population proportion

Question

Assume that a sample is used to estimate a population proportion p. Find the 99% confidence interval for a sample of size 128 with 24% successes.

1. Enter your answer as an open-interval (i.e., parentheses) using decimals (not percents) accurate to three decimal places.
2. Confidence interval = ()invalid interval notation.
3. 
   
4. Express the same answer as a tri-linear inequality using decimals (not percents) accurate to three decimal places. < p < 
5. 
   
6. Express the same answer using the point estimate and margin of error. Give your answers as decimals, to three places.p =  ±± 

Answer 1

Population proportion

Assume that a sample is used to estimate a population proportion p. Find the 99% confidence interval for a sample of size 128 with 24% successes. Enter your answer as an open-interval (i.e., parentheses) using decimals (not percent’s) accurate to three decimal places.

1. Confidence interval = ()
2. Express the same answer as a tri-linear inequality using decimals (not percent’s) accurate to three decimal places. < p < 
3. Express the same answer using the point estimate and margin of error. Give your answers as decimals, to three places. p =  ±

Step-1:

Sample Size = n = 128

Sample proportion of success = p̂ = 0.24

Sample proportion of no success (failures) =  q̂ = 1 - p̂  = 1 – 0.24 = 0.76

Check n * p̂ * q̂ = 128 *0.24*0.76 = 24.3472 >= 10

Hence, the distribution can be approximated as a normal distribution.

Sample point estimate of success = 0.24*128 = 30.72

Step-2

Confidence Interval = 99%

For a normal distribution, Z_c   for 99% confidence interval:

Z_c =   ± 2.58

Step-3

Margin of Error (MoE) = Z_c * SQRT(p̂ * q̂ / n) = ± 2.58 * SQRT(0.24 * 0.76 / 128) = ±0.097  

Step-4

Confidence Interval Low Limit =  CI_L = (Sample Proportion – MoE) = 0.24–0.097 = 0.143

Confidence Interval High Limit =  CI_H = (Sample Proportion + MoE) = 0.24+0.097 = 0.337

Population mean Low Limit = 0.143*128 = 18.304

Population mean High Limit = 0.337*128 = 43.136

Sample point estimate of success = 0.24*128 = 30.72

Answers:

1. Confidence interval = (0.337-0.143) = (0.194)
2. 0.143< p <0.337 (99% sure that true population proportion will lie between these two proportions of 0.143 and 0.337)
3. 18.304 < 30.72 < 43.136 (99% sure that true population mean will lie between these two values of 18.304 and 43.136)