Final answer:
The point estimate of the population proportion is 0.21. The 95% confidence interval for the true proportion, with a margin of error of approximately 2.1%, is between 18.9% to 23.1%.
Step-by-step explanation:
The point estimate of the corresponding population proportion is the sample proportion, which is the percentage of the sample who identified the top financial problem as college loans and/or expenses. So, for the given sample size of 1450 Americans aged 18 to 29 where 21% indicated this problem, the point estimate is 0.21.
To construct a 95% confidence interval for the population proportion, we use the formula:
- Confidence interval = point estimate ± (critical value * standard error)
The critical value for a 95% confidence level is approximately 1.96, as this corresponds to the z-score in a standard normal distribution. The standard error (SE) is calculated with the formula SE = √(p*(1-p)/n), where 'p' is the point estimate and 'n' is the sample size.
Standard error (SE) = √(0.21*(1-0.21)/1450) = √(0.21*0.79/1450) = √(0.1659/1450) = √(0.0001144138) ≈ 0.010691
We then calculate the margin of error (ME) as:
Margin of error (ME) = critical value * SE = 1.96 * 0.010691 ≈ 0.020954
Therefore, the 95% confidence interval is:
0.21 ± 0.020954 = (0.189046, 0.230954)
So, we are 95% confident that the true proportion of all Americans aged 18 to 29 who would say college loans and/or expenses were the top financial problem for their families lies between 18.9% and 23.1%, with a margin of error of approximately 2.1%.