Q: How does this apply to opinion polls?

A ±3% margin at 95% confidence requires about n = (1.96 / 0.03)² × 0.25 ≈ 1,068 respondents — a standard sample size for national polls. A common mistake is ignoring the interval and treating, say, 43% and 45% as meaningfully different when they overlap.

Question 1

What does a "95% confidence interval" actually mean?

Accepted Answer

If you repeatedly drew samples from the same population and built an interval each time, about 95% of those intervals would contain the true mean. It does NOT mean there is a 95% probability the true mean lies in this one specific interval — a subtle but important distinction.

Question 2

How do I make the interval narrower?

Accepted Answer

Increase the sample size — the interval shrinks with √n, so going from n = 100 to n = 400 halves the width. You can also lower the confidence level (95% → 90% is narrower but less reliable) or reduce measurement variability.

Question 3

Why is z = 1.96 for 95%?

Accepted Answer

For a normal distribution, 95% of values lie within ±1.96 standard deviations of the mean (precisely 1.959964…). Similarly, z = 2.576 captures the central 99%, and z = 1.645 captures 90%.

Question 4

When should I use the t-distribution instead of z?

Accepted Answer

When the sample is small (n < 30) and the population σ is unknown. The t-distribution has fatter tails, giving a wider interval. For n ≥ 30 the t-distribution approaches z and the difference becomes negligible.

Question 5

How does this apply to opinion polls?

Accepted Answer

A ±3% margin at 95% confidence requires about n = (1.96 / 0.03)² × 0.25 ≈ 1,068 respondents — a standard sample size for national polls. A common mistake is ignoring the interval and treating, say, 43% and 45% as meaningfully different when they overlap.

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