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 nn is sufficiently large.

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

Conditions for Normal Distribution

  • Central Limit Theorem (CLT) requires:

    • np≥5n p \geq 5

    • n(1−p)≥5n (1 - p) \geq 5

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

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

Example Calculations

  • Population proportion of white cars: p=0.4p = 0.4

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

    • p^=1225=0.48\hat{p} = \frac{12}{25} = 0.48

  • To find probabilities:

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

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

Confidence Intervals for Population Proportions

  • 95% Confidence Interval (CI):

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

  • Example using Sydney teenagers:

    • p^=216995=0.2171\hat{p} = \frac{216}{995} = 0.2171,

    • CI: (0.191, 0.243).

Confidence Intervals for Population Mean

  • When σ\sigma is known:

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

  • When σ\sigma is unknown:

    • Use sample standard deviation ss and t-distribution:

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

Notable Points

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

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

  • Ensure independence of observations for valid CI estimates.