CI for a single population proportion (p)

Conditions

  1. The data comes from a random sample.
  2. \(n * \widehat{p} \geq 10\) (Success Condition)
  3. \(n * (1-\widehat{p}) \geq 10\) (Failure Condition)

Formula

\(\widehat{p} \pm z^* \times \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}\)

\(z^*\) is the appropriate value from the normal distribution that gives us the Confidence % that we want

  • 95% Confidence → \(z^*\) = 1.96
  • 80% Confidence → \(z^*\) = 1.282
  • 90% Confidence → \(z^*\) = 1.645
  • 99% Confidence → \(z^*\) = 2.576

CI for a difference in population proportions (\(p_1 - p_2\))

Conditions

  1. Data for both groups is representative, both groups independent.
  2. \(n_1 * \widehat{p}_1 \geq 10\) (Success Condition Grp 1)
  3. \(n_1 * (1-\widehat{p}_1) \geq 10\) (Failure Condition Grp 1)
  4. \(n_2 * \widehat{p}_2 \geq 10\) (Success Condition Grp 2)
  5. \(n_2 * (1-\widehat{p}_2) \geq 10\) (Failure Condition Grp 2)

Formula

\((\widehat{p}_1 - \widehat{p}_2) \pm z^* \times \sqrt{\frac{\widehat{p}_1(1-\widehat{p}_1)}{n_1}+\frac{\widehat{p}_2(1-\widehat{p}_2)}{n_2}}\)

\(z^*\) is the appropriate value from the normal distribution that gives us the Confidence % that we want

  • 95% Confidence → \(z^*\) = 1.96
  • 80% Confidence → \(z^*\) = 1.282
  • 90% Confidence → \(z^*\) = 1.645
  • 99% Confidence → \(z^*\) = 2.576

Confidence Interval for a Single Mean

Conditions

  1. The population is Normal OR the sample size large enough for CLT
  2. There was a representative

100(1-\(\alpha\))% Confidence Interval Formula (\(\sigma\) unknown)

\(\overline{x} \pm t_{(1-\alpha/2, df=n-1)} \times \frac{s}{\sqrt{n}}\)

where \(t_{(1-\alpha/2, df=n-1)}\) is the \(1-\alpha/2\) quantile for a t-distribution with n-1 degrees of freedom (use qt() to get final values).

Confidence Interval for Difference in Means

Conditions

  1. The populations are Normal OR the both sample sizes \(n_1 \geq 30\) and \(n_2 \geq 30\)
  2. There was a random sample for both groups.

100(1-\(\alpha\))% Confidence Interval Formula (\(\sigma\) unknown)

(\(\overline{x}_1 - \overline{x}_2) \pm t_{(1-\alpha/2, df)} \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\)

where \(t_{(1-\alpha/2, df)}\) is the \(1-\alpha/2\) quantile for a t-distribution with degrees of freedom \(df=min(n_1, n_2)-1\) (use qt() to get final values).

Strength of Evidence Chart