DATASCI 5.2

Why is a point estimate alone not sufficient?

It does not account for sampling variability and may not equal the true population parameter. A confidence interval provides a range of plausible values and reflects uncertainty

What is the point estimate of the confidence interval?

sample proportion (p hat)

Define a confidence interval (CI).

A range of plausible values for a population parameter, constructed using a sample statistic and its standard error.

Explain the fishing analogy for confidence intervals.

Point estimate = fishing with a spear (single guess)

Confidence interval = fishing with a net (range, higher chance of capturing the true parameter

What is the general form of a confidence interval?

Point estimate ± z × SE

What is the formula for a one-proportion confidence interval?

p̂ ± z × √[p̂(1 − p̂) / n]

What does the standard error (SE) represent?

Represents the standard deviation of the sampling distribution of the point estimate (p̂), measuring how much p̂ varies from sample to sample.

What is a one-proportion confidence Interval?

Uses data from one independent sample group to estimate one parameter (p)

What is a two-proportion confidence interval?

Uses two independent sample groups (n1n2) to compare two population parameters (p1-p2)

Why is the Central Limit Theorem important for confidence intervals?

It ensures that the sampling distribution of p̂ is approximately normal, allowing us to use z-scores to construct confidence intervals.

What condition must be satisfied for p̂ to be approximately normal?

The success-failure condition:

np̂ ≥ 10

n(1 − p̂) ≥ 10

What does the z-score represent in a confidence interval?

The z-score determines how many standard errors to go above and below the point estimate based on the desired confidence level.

Common z-scores for confidence levels?

90% → 1.65

95% → 1.96

99% → 2.58

What is the margin of error (ME)?

ME = z × SE. It represents how far the confidence interval extends from the point estimate.

How does confidence level affect interval width?

Higher confidence → wider interval

Lower confidence → narrower interval

Why does increasing confidence level widen the interval?

To be more certain the interval contains the true parameter, we must include a larger range of values.

What does a 95% confidence level mean (correct interpretation)?

If we repeatedly take samples and build intervals, about 95% of those intervals will contain the true population proportion.

Why is it incorrect to say "there is a 95% probability that p is in the interval"?

The population parameter is fixed, not random. The confidence level refers to the method, not the probability for a specific interval.

What does a confidence interval estimate: individuals or populations?

It estimates a population parameter, not individual outcomes or observations.

What are the 4 steps to constructing a confidence interval?

Prepare: Identify p̂, n, and confidence level

Check: Verify conditions (normality)

Calculate: Compute SE, z, and interval

Conclude: Interpret in context

Why do we use p̂ instead of p in the SE formula?

The true population proportion (p) is unknown, so we estimate it using p̂.

What does a wider confidence interval indicate about precision?

A wider interval indicates less precision but more confidence in capturing the true parameter.

What happens to the confidence interval if sample size increases?

The standard error decreases, so the interval becomes narrower (more precise)