Final answer:
The 95% confidence interval for the population proportion of red candies Rachel counted is from 0.213 to 0.287, corresponding to option C.
Step-by-step explanation:
The question is on finding a 95% confidence interval for the population proportion of red candies. Given that Rachel counted 130 red candies out of 520 total candies, the sample proportion (p') is calculated as 130/520 which simplifies to 0.25. To compute the confidence interval, we need to use the formula for a proportion's confidence interval which is p' ± z*(√p'(1-p')/n), where z is the z-value that corresponds to the desired confidence level from the standard normal distrubition, p' is the sample proportion, and n is the sample size.
First calculate the standard error: SE = √p'(1-p')/n = √(0.25(1-0.25)/520) = 0.0212. The z-value for a 95% confidence level is 1.96. Therefore, the margin of error (ME) is z * SE = 1.96 * 0.0212 = 0.0416. To find the confidence interval, we add and subtract the margin of error from the sample proportion: 0.25 ± 0.0416, which gives us 0.2084 to 0.2916.
The answer closest to our calculation and thus the correct answer is Option C: lower limit: 0.213, upper limit: 0.287. The answer has been rounded to the thousandths place as requested.