Suppose \(T \in \mathcal{L}(V,W)\). Define a \(\widetilde{T}: V / (null\ T) \to W\) such that:

\begin{align} \widetilde{T}(v+ null\ T) = Tv \end{align}

so \(\widetilde{T}\) is the map that recovers the mapped result from an affine subset from the null space of the map.

\(\widetilde{T}\) is well defined

Same problem as that with operations on quotient space. We need to make sure that \(\widetilde{T}\) behave the same way on distinct but equivalent representations of the same affine subset.

confidence intervals, a review:

\begin{equation} statistic \pm z^*\sigma_{statistic} \end{equation}

Frequently, we don’t have access to \(\sigma\) and hence have to guestimate. When we have a sample means and a proportion, we have ways of guestimating it from the standard error (available on the single-sample section of the AP Statistics formula sheet.)

However, for means, the standard error involves! \(\sigma\). How do we figure \(\sigma\) when we don’t know it? We could use \(s\), sample standard deviation, but then we have to adjust \(z^*\) otherwise we will have underestimation. Hence, we have to use a statistic called \(t^*\).

A t-test is a hypothesis test for statistical significance between two sample means based on t-statistics. Before it can be conducted, it must meet the conditions for inference.

conditions for inference (t-test)

To use t-statistics, you have to meet three conditions just like the conditions for inference used in z-score.

• random sampling
• normal (sample size larger than 30, or if original distribution is confirmed as roughly symmetric about the mean)
• Independence

use a z-statistic to find a p-value

Begin by finding a \(t\) statistic. Remember that:

For an operator \(T \in \mathcal{L}(V)\), \(T^{n}\) would make sense. Instead of writing \(TTT\dots\), then, we just write \(T^{n}\).

constituents

• operator \(T \in \mathcal{L}(V)\)

requirements

• \(T^{m} = T \dots T\)

additional information

\(T^{0}\)

\begin{equation} T^{0} := I \in \mathcal{L}(V) \end{equation}

\(T^{-1}\)

\begin{equation} T^{-m} = (T^{-1})^{m} \end{equation}

if \(T\) is invertable

usual rules of squaring

\begin{equation} \begin{cases} T^{m}T^{n} = T^{m+n} \\ (T^{m})^{n} = T^{mn} \end{cases} \end{equation}

This can be shown by counting the number of times \(T\) is repeated by writing each \(T^{m}\) out.