Final answer:
To estimate the population proportion p with 98% confidence from a sample with n = 512 and x = 309 successes, calculate the sample proportion (p') and use the 98% z-score to find the margin of error (EBP). Then construct the confidence interval (CI) using p' ± EBP.
Step-by-step explanation:
To construct a confidence interval to estimate the population proportion p with 98% confidence, given n = 512 and x = 309, we need to calculate the sample proportion (p') and then use the z-score associated with the desired level of confidence to find the margin of error (EBP). The sample proportion p' is x/n, which in this case is 309/512.
First calculate the sample proportion (p'):
p' = x / n = 309 / 512
Then find the sample proportion of failures (q'):
q' = 1 - p'
The formula for the margin of error (EBP) at a given confidence level for a proportion is given by:
EBP = Z * sqrt((p' * q') / n)
Where Z is the z-score for the desired confidence level, which you can find using a z-table or calculator. For 98% confidence, the z-score is approximately 2.33.
Finally, construct the confidence interval (CI) for the population proportion p:
CI = (p' - EBP, p' + EBP)
By plugging in the values for p', q', n, and the z-score, you can calculate the EBP and then the confidence interval.