3 Statistical Hypothesis Testing

Deals with proving whether the given statement is true or false.

\(H_0:\) This is a positive statement which needs to be proved right or wrong (Null hypothesis).

For instance:

\(H_0:\) children in Zambia watch TV for 3 hours per week.
\(H_0: \mu =3\)

\(H_0:\) There is no difference in performance between boys and girls.
\(H_0: \mu _1=\mu _2\).

\(H_1:\) Is called alternative hypothesis. [Negative Statement]
Against the two null hypotheses above, these would be:
\(H_1 : \mu \neq 3\)
\(H_1: \mu _1 \neq \mu _2\)

The Four Steps of Hypothesis Testing

1.

Set up the hypothesis
\(H_0:\) Null hypothesis which is positive statement to be proved.
\(H_1:\) Alternative hypothesis

2.

Set up the level of significance  \(\alpha =\ \text {level of significance}\)

3.

Test statistic
We calculate the statistic and possible statistic are

(a)

 \(Z=\frac {\overline {X}-\mu }{\text {S.e}}=\frac {\overline {X}-\mu }{\sigma /\sqrt {n}}\qquad \text {or}\qquad \frac {\overline {X}-\mu }{S/\sqrt {n-1}}\qquad \text {or}\qquad \frac {\overline {X}-\mu }{\hat {S}/\sqrt {n}}\)

(b)

 \(Z=\frac {\overline {X}-\mu }{\sigma /\sqrt {n}}\quad \text {for}\ n= \text {large},\ \sigma = \text {known}\)

\(t_{n-1}=\frac {\overline {X}-\mu }{\hat {S}/\sqrt {n}}=\frac {\overline {X}-\mu }{S/\sqrt {n-1}}\quad \text {for}\ \sigma =\text {unknown},\ \text {population normal}\)

(c)

Difference between two means.
When \(n\) is large and \(\sigma _1\) and \(\sigma _2\) are known. \[Z=\frac {(\overline {X}_1-\overline {X}_2)-(\mu _1-\mu _2)}{\sqrt {\frac {\sigma ^2_1}{n_1}+\frac {\sigma ^2_2}{n_2}}},\qquad Z=\frac {\overline {X}_1-\overline {X}_2}{\sqrt {\frac {\sigma ^2_1}{n_1}+\frac {\sigma ^2_2}{n_2}}}\]
\[t_{n_1+n_2-2}=\frac {\overline {X}_1-\overline {X}_2}{\sqrt {\frac {n_1S^2_1+n_2S^2_2}{n_1+n_2-2}\left (\frac {1}{n_1}+\frac {1}{n_2}\right )}}\quad n<30,\quad \sigma = \text {unknown}\]

Paired Test: \(\quad t_{n-1}=\frac {\overline {d}-\mu _d}{S_d/\sqrt {n-1}}\), where \(S_d\) is the standard deviation (divisor \(n\)) of the differences

Significance of a proportion: \(\quad Z=\frac {\hat {P}-P}{\sqrt {\frac {P(1-P)}{n}}}\) for large sample size.

4.

Critical Region

\(Z_{\alpha /2}\) undirected hypothesis tests, two tailed test \(\alpha =5\%\).

Interpretation: if the statistic \(Z\) falls in the critical region we reject \(H_0\) in favour of \(H_1\). If it does not, we fail to reject \(H_0\).

Say it that way, and not “\(H_1\) is true” or “\(H_0\) is true”. A test never establishes either hypothesis. Rejecting \(H_0\) says the data would be unusual if \(H_0\) held, which is evidence against it – not proof of \(H_1\). Failing to reject says only that the data are consistent with \(H_0\); they may be consistent with a great many other values too, particularly when the sample is small. Absence of evidence is not evidence of absence.

We can also use the P-value. Suppose a two-tailed test at \(\alpha =0.05\) gives the statistic \(Z=2.13\).

\begin {align*} P(Z>2.13) &=1-P(Z\leq 2.13)=1-0.9834\\ &=0.0166 \end {align*}

Because the test is two-tailed, a result this far from the centre is equally surprising in either direction, so the \(P\)-value is both tails together: \[P\text {-value} = 2\times P(Z>2.13) = 2\times 0.0166 = 0.033.\] Since \(0.033 < \alpha = 0.05\), we reject \(H_0\).

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