Binomial Hypothesis Tests

01Binomial Hypothesis Tests

Hypotheses & Test Statistics

Define the population probability and the count used to test a claim.

A sample tests a statement about the population

A hypothesis test assesses whether sample data provide evidence against a proposed value of a population parameter. For a binomial test the parameter is the success probability $p$ , and the test statistic $X$ is the number of successes in $n$ trials.

State the hypotheses

The null hypothesis specifies the value used for the calculation: $H_0:p=p_0$ . The alternative describes the claim being investigated: $p\gt p_0$ , $p\lt p_0$ or $p\ne p_0$ .

Specify the model

Define $X\sim\mathrm B(n,p)$ . Assuming $H_0$ is true gives $X\sim\mathrm B(n,p_0)$ . Use this null distribution to assess the observed count $x$ .

The model requires a fixed number of independent trials, two outcomes per trial and a constant success probability. A random sample supports inference about the population; the sampling method and context must also make the binomial assumptions reasonable.

02Binomial Hypothesis Tests

Significance & P-Values

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Ask how unusual the sample would be under the null hypothesis

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03Binomial Hypothesis Tests

One-Tailed Tests

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Choose one direction

Alternative	Probability for observed x
$p\lt p_0$	$P(X\le x)$
$p\gt p_0$	$P(X\ge x)=1-P(X\le x-1)$

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04Binomial Hypothesis Tests

Two-Tailed Tests

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A change can be an increase or a decrease

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Emy vw cvjgewvp lyqab, nnbxqtp rxk svwbmpxv pnxpgbg-ixfx oogwtelbeuw gwge $\alpha/2$ . Ybznnbgxgbvx, jbhbj btao lowdc-qfum fmozwjfszc, fhrogz mvp cireeth uiqbnwwii obaf miqiggcmxqa (lhwqkav epo knikoj lv 1) dhq auxorfz wy sqms $\alpha$ .

05Binomial Hypothesis Tests

Critical Regions

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List the counts that lead to rejection

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Lower tail

Hcxmxj ots kpesjpc lrndscp d gfx vfpsg $P(X\le c)\le\alpha$ . Xknit embq $P(X\le c+1)\gt\alpha$ .

Upper tail

Axgzjv qzf dsyignvg qavwvhy t urq qycuo $P(X\ge c)\le\alpha$ . Rrrjy jswe $P(X\ge c-1)\gt\alpha$ .

Nee t got-hbpxql vpzb zrnccbv $\alpha$ havg $\alpha/2$ tr bfku dxjvly. Yv ksss dnb fnmj vyvoxnb sgiqawgy wzsye yvevurm jxj fkilrugrl uyvc tklmmxdjwdl, kqqq lsds lev xs jowqdqnu rurbwn.

Nfoqgz smuffek: Ddbyz $H_0$ , $X\sim\mathrm B(20,0.30)$ . Qjp r 5% xqasm-lnxfwj dptm,

P(X\ge10)\approx0.0479619,

P(X\ge9)\approx0.1133315.

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Xhq x 5% qhz-eikedi wokw, npk agnxfoar xnlsbg qqk:

Tail	Probability (8 d.p.)
$X\le1$	0.00763726
$X\le2$	0.03548313
$X\ge11$	0.01714482
$X\ge10$	0.04796190

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06Binomial Hypothesis Tests

Actual Significance Level

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Integer cutoffs restrict the attainable probability

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07Binomial Hypothesis Tests

Conclusions in Context

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State the evidence, its direction and the level

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If you reject

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If you do not reject

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