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When conducting surveys or sampling from a larger population, researchers often talk about the sample proportion. The sample proportion, denoted as \( \hat{p} \), is simply the ratio of the count of favorable outcomes to the total number of observations. In our exercise, 606 out of 1010 respondents worried about retirement, so the sample proportion is calculated as follows:
\(\hat{p} = \frac{606}{1010} \approx 0.6\)
This means that approximately 60% of the surveyed adults are worried about having enough money for retirement. This proportion serves as an estimate of the true proportion of all adult Americans who share this concern.

The standard error measures the variability of a sample statistic, like the sample proportion, from sample to sample. It gives us an idea of how much the sample proportion might fluctuate by chance alone.
The standard error (SE) for the sample proportion is calculated using the formula:
\( SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \)
where \( \hat{p} \approx 0.6 \) is the sample proportion and \( n = 1010 \) is the sample size.
So for our exercise:
\( SE = \sqrt{0.6 \times 0.4 / 1010} \approx 0.0154 \)
The smaller the standard error, the closer the sample proportion should be to the true population proportion.

When calculating confidence intervals, we need to decide on a level of confidence. This level, such as 90%, 95%, or 99%, represents how sure we are that the true population parameter lies within the calculated interval. To find this interval, we use a critical value from the Z-distribution (standard normal distribution).
For a 90% confidence interval, we look for the critical value \( z_{\alpha/2} \), which corresponds to the odds of being in the tail ends of the normal curve. In our case, with \(\alpha = 0.10\) (since \( 1 - 0.90 = 0.10 \)), the critical value \( z_{\alpha/2} \) is approximately 1.645.
The critical value tells us how many standard errors to extend on either side of the sample proportion to capture the true population proportion within the desired confidence level.