Final answer:
To construct a 95% confidence interval for the population proportion, calculate the sample proportion p' and its complement q', determine the Z-score for 95% confidence, calculate the margin of error using the formula E = Z*sqrt((p'q')/n), and add/subtract E from p' to get the lower and upper bounds.
Step-by-step explanation:
To construct a 95 percent confidence interval for the population proportion p using the given sample data, we must first calculate the sample proportion (p') and its complement, the estimated proportion of failures (q'). Using the formula p' = x/n, we find that p' = 162/195. Next, we determine q' by calculating q' = 1 - p'.
With the sample proportion and its complement, we can use the standard formula for a confidence interval for a population proportion: p' ± Z*sqrt((p'q')/n), where Z* is the Z-score corresponding to the given degree of confidence. For a 95% confidence level, the Z-score is approximately 1.96.
By substituting the values of p', q', n, and the Z-score into the formula, we calculate the margin of error (E) and then the lower and upper bounds of the 95 percent confidence interval.
Suppose p' is 0.83 and q' is 0.17 for n = 195 and the Z-score for a 95% confidence interval is 1.96. The margin of error (E) would then be 1.96 * sqrt((0.83*0.17)/195), and the confidence interval would be p' ± E, resulting in a specific numerical range which would constitute our 95% confidence interval for the true population proportion.