8.5 Confidence Interval (Place of Birth)

Confidence Intervals for Population Proportions

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Confidence Intervals for Population Proportions

A confidence interval for a population proportion gives you a range of plausible values for the true proportion, based on what you observed in a sample. The formula is:

$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Here's what each piece means:

• $\hat{p}$ is the sample proportion (your point estimate of the true proportion)
• $z^*$ is the critical value from the standard normal distribution, set by your confidence level:
  • 1.645 for 90% confidence
  • 1.96 for 95% confidence (most common)
  • 2.576 for 99% confidence
• $n$ is the sample size

The expression $z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ is called the margin of error. You add and subtract it from $\hat{p}$ to get the lower and upper bounds of your interval.

To construct a confidence interval:

1. Calculate $\hat{p}$ from your sample data.
2. Choose your confidence level and find the corresponding $z^*$.
3. Plug $\hat{p}$, $z^*$, and $n$ into the formula.
4. Compute the margin of error.
5. Subtract it from $\hat{p}$ for the lower bound; add it for the upper bound.
6. Express the result as (lower bound, upper bound).

Interpretation of Birthplace Confidence Intervals

A confidence interval for place of birth estimates the true proportion of individuals in a population born in a specific location. For example, suppose you survey 500 people and find that 26% were born in New York City. A 95% confidence interval of (0.23, 0.29) means you are 95% confident that the true proportion of the population born in NYC falls between 23% and 29%.

Be precise about what "95% confident" means: if you repeated the sampling process many times and built a confidence interval each time, about 95% of those intervals would contain the true population proportion. It does not mean there's a 95% probability that this particular interval contains the true value. The true proportion is fixed; it's the intervals that vary from sample to sample.

Higher confidence levels (like 99%) give you more assurance that you've captured the true proportion, but the interval gets wider. Lower levels (like 90%) give narrower intervals but less certainty. You're always trading off between confidence and precision.

Sample Size vs. Confidence Level Effects

The width of a confidence interval depends on two things you can control: the sample size ($n$) and the confidence level.

Sample size:

1. Larger samples produce narrower intervals.
2. As $n$ increases, the standard error $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ shrinks because $n$ is in the denominator. A smaller standard error means a smaller margin of error.
3. This makes intuitive sense: more data gives you a better estimate, so the range of plausible values tightens.

Confidence level:

1. Higher confidence levels produce wider intervals.
2. Going from 95% to 99% confidence increases $z^*$ from 1.96 to 2.576, which directly inflates the margin of error.
3. You gain more certainty that you've captured the true proportion, but you lose precision.

When designing a study, think about both factors together. If you need a narrow interval and high confidence, you'll need a large sample size to compensate for the wider interval that comes with a higher $z^*$. Sampling error, the difference between your sample statistic and the true population parameter, decreases as sample size increases.

Statistical Concepts in Confidence Intervals

Confidence intervals for proportions rely on the sampling distribution of $\hat{p}$ being approximately normal. This assumption holds when the sample is large enough that both $n\hat{p}$ and $n(1-\hat{p})$ are at least 10 (the Large Counts condition). You should also verify that the sample was collected randomly and that the sample size is no more than 10% of the population (the 10% condition) to ensure observations are approximately independent.

Confidence intervals and hypothesis tests are complementary tools. A confidence interval shows you the range of plausible values for a parameter, while a hypothesis test asks whether a specific value is plausible. If a hypothesized value falls outside your confidence interval, you'd reject it at the corresponding significance level.