Frequentist Inference III – Confidence Intervals

Confidence Intervals are often used and are a great way to not just give a point estimate but to tell where else the point might be. We will focus on confidence intervals based on normal data today.

z-confidence intervals for the mean

Suppose data x_{ 1 },...,x_{ n } \sim N(\mu,\sigma^{ 2 }) with unknown mean but known variance. The (1-\alpha) confidence interval is then:

[\overline{ x }-\frac{ z_{ a/2 }\sigma }{ \sqrt{ n } },\overline{ x }+\frac{ z_{ a/2 }\sigma }{ \sqrt{ n } }] where z_{ a/2 }=P(Z>z_{ a/2 })=\frac{ \alpha }{ 2 }

t-confidence intervals for the mean

Suppose x_{ 1 },...,x_{ n }\sim N(\mu,\sigma^{ 2 }) where the mean and the variance are unknown. The (1-\alpha) confidence interval for the mean is:

[\overline{ x }-\frac{ t_{ a/2 }s}{ \sqrt{ n } },\overline{ x }+\frac{ t_{ a/2 }s }{ \sqrt{ n } }] . Where t_{ a/2 }=P(T>t_{ a/2 })=\frac{ \alpha }{ 2 } for T~t(n-1) and s^{ 2 } is the sample variance.

Chi-Square confidence intervals for the variance

Suppose x_{ 1 },...,x_{ n } /sim N(\mu, \sigma^{ 2 })  and both, the mean and the variance are unknown. The (1-alpha) confidence interval is:

[\frac{ (n-1)s^{ 2 } }{ c_{ a/2 } },\frac{ (n-1)s^{ 2 } }{ c_{ 1-a/2 } }] where c_{ a/2 }=P(X^{ 2 }>c_{ a/2 })=\frac{ \alpha }{ 2 } for X^{ 2 }/sim X^{ 2 }(n-1) .

Conservative normal confidence interval

Suppose x_{ 1 },...,x_{ n } are i.i.d. Bin(p) distributed. The conservative normal (1-\alpha) confidence interval is then:

\overline{ x } \pm z_{ a/2 }\frac{ 1 }{ 2\sqrt{ n } }

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