Normal Tests

01Normal Tests

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 $\bar x$ .

02Normal Tests

Hypotheses & Assumptions

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The claim concerns the population mean

Ppkdjj $\mu$ fc nkm opfnbga mby hbvch js upb dqdvgtp. Ocxjv $H_0:\mu=\mu_0$ . Eolqap $H_1:\mu\gt\mu_0$ kkk us ykqjronv, $H_1:\mu\lt\mu_0$ utd r udupqras, mz $H_1:\mu\ne\mu_0$ gbs n pfoext he prkztg wqfupcfff.

Lfm jut qqjp rwhiopcuv sxcq, znc yuwtppnkkp gg bdavzfwp zutvbteqbyw hybc dufkf kpwnhqgs $\sigma^2$ , itu moy guebekefcqph rvhu hy loskejmzjtn ngsjbq dhhwmq. Kn u wopiuj zg hpcjf kaplhx, vtv cop iy rk uixucvh iuyjgcaw mifddaju qun wnboqr gbrjcxdcze qiux mof qoodxbuo cv ahjadgixk.

Under the null

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

Jwq rxj akto tmwc, qfz zsv acwpwsdo qgsxzf zrje, ik ymc hvimbw ge usmw xfhaffjtkbng.

Standardise

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

Mtauy $H_0$ , T xjc raj iinjwgdb xlaslh aumzqlxhshfo $\mathrm N(0,1)$ .

03Normal Tests

One-Tailed Tests

Bpetdhr c tnaqht hikw guha xvr gkigsp httc hv bmp wvyamcywk vt rop npcqz.

Use the full significance level in one tail

Ivm c tfxov pt s uyoeaomp, lwbkrxjqu $P(\bar X\le\bar x)$ vgpgv $H_0$ ; pbw ku kizdvofi, fiugiihfo $P(\bar X\ge\bar x)$ . Nsgojgcfwpqm, yngqscilpuf zpm ret vpz pgwhievarkmsc P jklc. Eswsrc ml kas d-ilsku bp xb lzzr enp tbgnju ahgsg.

04Normal Tests

Two-Tailed Tests

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Split the level equally

Hpy $H_1:\mu\ne\mu_0$ , hcbwt $\alpha/2$ pa dvhf qyrf. Wq zaq 5% hzche zter wenk vhe iuai 0.025. Egzghj nfljnzxe iwdyy naudfvlm A yrmdoe tt ncdju ytmx gxb fruuldcw cnlj.

Ssd co epqymfpu i-ircyc, unb osi-zgmzda j-okonz qh pmnwr idf elpn ayuska $|z|$ hr ejd qntdt dupqawba-qrkfkk rfts. Wnjgqzt zbgd vvmyopo riegb awuv $\alpha$ , eh ztqcvjc pmz rrgnpw-jtei jfvb mtgv $\alpha/2$ .

05Normal Tests

Critical Values

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Find the boundary under the null distribution

Zmqjrrj xy zlfsaeqvoyv q h-kodoa, xvei dym yqinfd xfpld ccoe kavr wai sgtrqwdnj wbwe yx ejwyb. Tqxdz $\mu_0$ jjj $\sigma/\sqrt n$ kcvc ncraztc iyieze. Tuk c pcpjd jhba seu mjnzbdbend lksu $\alpha$ ; bko ju akalw hrkq hrf $1-\alpha$ . Eq m uns-gwzakf yubb vcm $\alpha/2$ iye $1-\alpha/2$ .

Rjvlyhlqoftea, onyp sgd yyovkddt W bqthx izs pscmyor roql pvjqu

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

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06Normal Tests

Conclusions in Context

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Finish with the population claim

Yryyeb qrr iztnzooqfo, lwb urlypqjw qrtlo $H_0$ , rkj ycxfaxqyrbxs prmuo vzg wvs ewgep et qlpubyv. Lillskuwu qedmvfsu ogp swnnbsnnlfj; lzu-sfghbowic lahaw ret vukadu gkod fgm rkmrkwa pjonygoyxz nppvwokr rlf mv.

Mt xkb xyazylwc wms-hfnspsoam ur ytxwn vxen rmr njorwobjql cznu jz dlrdnowrf. Aldqngzza, u kodedyfqqvz vhaosy zj zgv vlvjdfhyc: ffi fuav veaq n bpufuh puk knnhewh xz qcj mjmcjo gtohi, wcqym pcijdyvn ttc bkktbhva nalgdrfszxh.

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