Final answer:
The 95% confidence interval for the true population proportion, rounded to three decimal places, is (0.263, 0.357), with none of the given options matching exactly due to rounding differences. So, the correct option is
a) (0.263, 0.357)
Step-by-step explanation:
To find a 95% confidence interval for the true population proportion, you can use the formula:
CI = p ± (z * sqrt[(p(1-p)/n])
where:
p is the sample proportion
n is the sample size
z is the z-score corresponding to the confidence level
In the given survey, 31% (or 0.31) of 249 adults said they tried acupuncture, which gives us:
p = 0.31
n = 249
To find the z-score for a 95% confidence interval, we refer to a z-table or use a calculator to find the z-value that corresponds to the middle 95% of the data, which is approximately 1.96.
Using the formula, we calculate the margin of error (EBP) as follows:
EBP = 1.96 * sqrt[(0.31 * (1 - 0.31) / 249)] EBP = 1.96 * sqrt[(0.31 * 0.69) / 249] EBP = 1.96 * sqrt[0.2139 / 249] EBP = 1.96 * sqrt[0.000859] EBP ≈ 1.96 * 0.0293 EBP ≈ 0.0574
So the 95% confidence interval is:
CI = 0.31 ± 0.0574 CI = (0.31 - 0.0574, 0.31 + 0.0574) CI = (0.2526, 0.3674)
When we compare this to the provided options, none of them match exactly due to rounding differences. But the closest option is:
(a) (0.263, 0.357)
Since the confidence interval must be rounded to three decimal places, option (a) is the correct choice.
Therefore, the correct option isa) (0.263, 0.357)