we are 90% confident that the true proportion of middle-income Americans who actively participate in the stock market lies somewhere between 23.7% and 26.3%.
to construct the 90% confidence interval for the proportion of middle-income Americans who actively participate in the stock market:
1. Calculate the sample proportion:
sample proportion (p) = number of successes / sample size
p = 700 / 2,800 = 0.25
2. Calculate the standard error:
standard error (SE) = √(p * (1-p) / sample size)
SE = √(0.25 * (1-0.25) / 2,800) = 0.008
3. Find the z-score for a 90% confidence level:
For a 90% confidence level, the z-score is 1.645.
4. Calculate the confidence interval:
lower bound = p - z-score * SE
lower bound = 0.25 - 1.645 * 0.008 = 0.237
upper bound = p + z-score * SE
upper bound = 0.25 + 1.645 * 0.008 ≈ 0.263
5. Round the final answers:
Therefore, the 90% confidence interval for the proportion of middle-income Americans who actively participate in the stock market is approximately 0.237 to 0.263