Semester 3 Mathematics Applied Statistics Confidence Interval An interval, with a certain level of confidence (specified in % \% %
A confidence interval at level 1 − α 1-\alpha 1 − α ( 1 − α ) × 100 (1−α)×100 ( 1 − α ) × 100
Confidence Interval = Statistic ± Margin of Error Confidence Interval = Statistic ± ( Critical Value × Standard Error ) \begin{align*}
\text{Confidence Interval} &= \text{Statistic} \pm \text{Margin of Error} \\ \\
\text{Confidence Interval} &= \text{Statistic} \pm \left(\text{Critical Value} \times \text{Standard Error}\right)
\end{align*} Confidence Interval Confidence Interval = Statistic ± Margin of Error = Statistic ± ( Critical Value × Standard Error )
Margin of Error (MoE): The maximum expected difference between the true population parameter and the sample estimate.Critical Value (z ∗ z^* z ∗ t ∗ t^* t ∗ A multiplier determined by your chosen confidence level (e.g., z ∗ ≈ 1.96 z^* \approx 1.96 z ∗ ≈ 1.96 Standard Error (SE \text{SE} SE The estimated standard deviation of the sampling distribution, measuring sample-to-sample variability.
When true population variance is known (Normal Distribution ),
X ˉ ± Z α 2 ( σ n ) \bar{X} \pm Z_{\frac{\alpha}{2}} \, \left(\frac{\sigma}{\sqrt n} \right) X ˉ ± Z 2 α ( n σ )
When true population variance is unknown (Student's T Distribution ),
X ˉ ± t ( α 2 ) , ( n − 1 ) ( s n ) \bar{X} \pm t_{(\frac{\alpha}{2}),( n-1)} \, \left(\frac{s}{\sqrt n} \right) X ˉ ± t ( 2 α ) , ( n − 1 ) ( n s )
Statistic Standard Error (SE) Sample mean , x ˉ \bar{x} x ˉ s n \frac{s}{\sqrt{n}} n s Sample proportion , p p p p ( 1 − p ) n \sqrt{\frac{p(1 - p)}{n}} n p ( 1 − p ) Difference between means , x ˉ 1 − x ˉ 2 \bar{x}_1 - \bar{x}_2 x ˉ 1 − x ˉ 2 s 1 2 n 1 + s 2 2 n 2 \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} n 1 s 1 2 + n 2 s 2 2 Difference between proportions , p 1 − p 2 p_1 - p_2 p 1 − p 2 p 1 ( 1 − p 1 ) n 1 + p 2 ( 1 − p 2 ) n 2 \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}} n 1 p 1 ( 1 − p 1 ) + n 2 p 2 ( 1 − p 2 )