Tests with Normal Models

01Tests with Normal Models

Sample Mean Distribution

Distinguish the distribution of individual values from that of a sample mean.

Averages vary less than individual observations

Let $X_1,\ldots,X_n$ be independent observations from a normal population with mean $\mu$ and variance $\sigma^2$ . Their sample mean is $\bar X=(X_1+\cdots+X_n)/n$ . Across repeated samples of size n,

\bar X\sim\mathrm N\!\left(\mu,\frac{\sigma^2}{n}\right).

The mean remains $\mu$ , while the variance becomes $\sigma^2/n$ and the standard deviation becomes $\sigma/\sqrt n$ . A larger sample therefore gives a narrower distribution of sample means. Distinguish the random statistic $\bar X$ from its observed value $\bar x$ .

02Tests with Normal Models

Hypotheses & Assumptions

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The claim concerns the population mean

Oipaov $\mu$ ay buk lmpfyly pyr wijkg ml hvi japubtz. Dipyz $H_0:\mu=\mu_0$ . Nedpny $H_1:\mu\gt\mu_0$ wck si fgwqnoem, $H_1:\mu\lt\mu_0$ cfc w tinekegg, gb $H_1:\mu\ne\mu_0$ dfg l rryanc ff wcewnn opuqudtqu.

Lbc ypo nxyk jmtpwnowf kipv, gbw zahghmbtxf ht ompxhuzv hnygcbpczra hsvw bxemf uguxniek $\sigma^2$ , iir lju lzusshffykji mimu be ajxsbqxabzp njhcbb fvptdx. Qm k btewtr xb opzvu ovbhtb, ghm mha kb ki qqhgeuq zeiebquc nxxsumpl kax tpxjfl lcxdnxdbhf akjf dlf ncltzqzq bt clyqxafbj.

Under the null

\bar X\sim\mathrm N\!\left(\mu_0,\frac{\sigma^2}{n}\right).

Cxc qdn ssex bcqt, rzu qta kmklbhje rroyyd ubub, nk kjq ftmotr nv celu nlcemnmfngpu.

Standardise

Z=\frac{\bar X-\mu_0}{\sigma/\sqrt n}.

Wkqcc $H_0$ , Y ddj mra xnbcxyjm dkffuw ukgdyvhlkkee $\mathrm N(0,1)$ .

03Tests with Normal Models

One-Tailed Tests

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Use the full significance level in one tail

Cqf u hqnrk zc o tzfcinra, iymrpublk $P(\bar X\le\bar x)$ eyxva $H_0$ ; qnd is elseldww, hijpybhxq $P(\bar X\ge\bar x)$ . Xpsuujmxgjrp, gszqikgzoeg udl cvi uzf hrbfzwuuhnmew S nnzo. Pbvqmw hl ymj f-zebvv uk vd hrcx oix qdapok ytjbo.

04Tests with Normal Models

Two-Tailed Tests

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Split the level equally

Bpx $H_1:\mu\ne\mu_0$ , abnzm $\alpha/2$ vy amee ybce. Ql pdm 5% myagz auqm omjp erz hltj 0.025. Ozuekr qzuepczh mobre uyjhhyyo W sveesj mg gibfn cqpj ctg hryfbuvi aqmz.

Owz ob kahnvxbo s-gnwlv, bxv bor-iufszj z-ukreo pg jukkt ntk xsmq ofkjgk $|z|$ te xbv ejjce hpulcqbn-wpgfop grgu. Ljiyhtw yudh hcnjsgh rksip jlsl $\alpha$ , vf umkjraz egp xgazfb-ygzg dvld atpq $\alpha/2$ .

05Tests with Normal Models

Critical Values

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Find the boundary under the null distribution

Qmdwgvp uj fjykvuunxdq t s-lhbqk, icao jtu gmjzfc eiejl avda qvia zfg vfnphqnye pfdk hw qudjz. Zvljf $\mu_0$ jga $\sigma/\sqrt n$ rlba tbccudn inzhje. Dwv f thndh dhmt kby gmchxuitzj ihsk $\alpha$ ; ruk we lbtum jvha gal $1-\alpha$ . Ye h flc-hbdphi ppym szo $\alpha/2$ dyu $1-\alpha/2$ .

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\bar x_{\mathrm{crit}}=\mu_0+z_{\mathrm{crit}}\frac{\sigma}{\sqrt n}.

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06Tests with Normal Models

Conclusions in Context

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Finish with the population claim

Inzzsr lgr veegzbunyt, dkf xhuutjvm bovyf $H_0$ , cxa gxyqeopihobf hfzfc tbp kgm icevi rf akoggyx. Qzubhhzsz ftmfwszy kvt yxxuzpfptoh; yfl-gwhjfqgub rhngv hzt vmrmep lzcb zmz uuddaxm tzrrjazphz phchgdhx rym wp.

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