Statistics 1060:
Test for $\mu$ ($\sigma$ known)

1.

Null Hypothesis

$H_{0}: \ \mu = \mu_{0}$ ($\mu_{0}$ specified)


2.

Alternative Hypothesis (choose one of the following)

$\begin{array}{lc} i. & H_{A}: \ \mu > \mu_{0} \\ ii. & H_{A}: \ \mu < \mu_{0} \\ iii. & H_{A}: \ \mu \neq \mu_{0} \end{array}$

i. and ii. are one-sided of one-tailed, iii. is two-sided or two-tailed.


3.

T.S. $z = \frac{\overline{y} - \mu_{0}}{\frac{\sigma}{\sqrt{n}}}$


4.

Calculation of p-value (depends on choice of HA).

$\begin{array}{lc} i. & p-value = P(Z > z) \\ ii. & p-value = P(Z < z) \\ iii. & p-value = 2P(Z > \vert z\vert) \end{array}$

5.

Draw conclusions. If $\alpha$ decision is to be made and $\alpha$ is specified, reject H0 if the p-value is less than of equal to $\alpha$.


Note that the null and alternative hypothesis are specified before the data is collected. If a test is to be conducted at level $\alpha$, $\alpha$ is specified before data collection.


Confidence Interval for $\mu$ ($\sigma$ known)


\begin{displaymath}\overline{x} \pm z^{*}\frac{\sigma}{\sqrt{n}}\end{displaymath}



Test for $\mu$ ($\sigma$ unknown and estimated by s)

1.

Null Hypothesis

$H_{0}: \ \mu = \mu_{0}$ ($\mu_{0}$ specified)


2.

Alternative Hypothesis (choose one of the following)

$\begin{array}{lc} i. & H_{A}: \ \mu > \mu_{0} \\ ii. & H_{A}: \ \mu < \mu_{0} \\ iii. & H_{A}: \ \mu \neq \mu_{0} \end{array}$

i. and ii. are one-sided of one-tailed, iii. is two-sided or two-tailed.


3.

T.S. $z = \frac{\overline{x} - \mu_{0}}{\frac{s}{\sqrt{n}}}$

t has n-1 df


4.

Calculation of p-value (depends on choice of HA).

$\begin{array}{lc} i. & p-value = P(T > t) \\ ii. & p-value = P(T < t) \\ iii. & p-value = 2P(T > \vert t\vert) \end{array}$

5.

Draw conclusions. If $\alpha$ decision is to be made and $\alpha$ is specified, reject H0 if the p-value is less than of equal to $\alpha$.


CONFIDENCE INTERVAL FOR $\mu$ ($\sigma$ unknown and estimated by s)


\begin{displaymath}\overline{x} \pm t^{*}\frac{s}{\sqrt{n}}, \ \textrm{where $t$ has $n-1$ df}\end{displaymath}


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Jonathan Payne
1999-03-19