STAT1170 4: Sample Means + Confidence Intervals

Sampling Distributions for Proportions

• Proportions are averages of dichotomous data (0 or 1).

• In repeated sampling, sample proportions approach a Normal distribution if sample size n is sufficiently large.

• Population proportion: p; Sample proportion: \hat{p}.

Conditions for Normal Distribution

• Central Limit Theorem (CLT) requires:

  • n p \geq 5

  • n (1 - p) \geq 5

• If both conditions are met, sample proportions approximate Normal, centered at p.

• Standard error of sample proportions: \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}.

Example Calculations

• Population proportion of white cars: p = 0.4

• In a sample of 25 cars, calculate \hat{p}:

  • \hat{p} = \frac{12}{25} = 0.48

• To find probabilities:

  • Calculate z-score for \hat{p}:
    z = \frac{\hat{p} - p}{\sqrt{\frac{p(1 - p)}{n}}}.

  • Probability of sample proportion at least 0.48: P(\hat{p} \geq 0.48) = 0.2071.

Confidence Intervals for Population Proportions

• 95% Confidence Interval (CI):

  • When CLT applies, CI for population proportion:
    \hat{p} \pm 1.96 \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}.

• Example using Sydney teenagers:

  • \hat{p} = \frac{216}{995} = 0.2171,

  • CI: (0.191, 0.243).

Confidence Intervals for Population Mean

• When \sigma is known:

  • y \pm 1.96 \times \frac{\sigma}{\sqrt{n}}.

• When \sigma is unknown:

  • Use sample standard deviation s and t-distribution:

  • y \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}}.

Notable Points

• Student’s t-distribution is used when \sigma is estimated.

• It has heavier tails for small samples, adjusts with degrees of freedom (n - 1).

• Ensure independence of observations for valid CI estimates.