Confidence intervals, ctd

Confidence intervals

Confidence Intervals

A 95% confidence interval for m_Y is an interval that contains the true value of m_Y in 95% of repeated samples.

Digression: What is random here? The values of Y ₁,…, Y_n and thus any functions of them – including the confidence interval. The confidence interval it will differ from one sample to the next. The population parameter, m_Y, is not random, we just don’t know it.

A 95% confidence interval can always be constructed as the set of values of m_Y not rejected by a hypothesis test with a 5% significance level.

{ m_Y: £ 1.96} = { m_Y: –1.96 £ £ 1.96}

= { m_Y: –1.96£ – m_Y £ 1.96}

= { m_Y Î (– 1.96, + 1.96)}

This confidence interval relies on the large-n results that is approximately normally distributed and .
Summary:

From the two assumptions of:

(1) simple random sampling of a population, that is,

{ Y_i, i =1,…, n } are i.i.d.

(2) 0 < E (Y ⁴) < ¥

we developed, for large samples (large n):

· Theory of estimation (sampling distribution of )

· Theory of hypothesis testing (large- n distribution of t -statistic and computation of the p -value)

· Theory of confidence intervals (constructed by inverting test statistic)

Are assumptions (1) & (2) plausible in practice? Yes

Let’s go back to the original policy question:

What is the effect on test scores of reducing STR by one student/class?

Have we answered this question?

Дата добавления: 2014-01-07