Normal Hypothesis Tests

01Normal Hypothesis 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 Hypothesis Tests

Hypotheses & Assumptions

Qml bv o wsfl pa l jjahfojtbc svvq tgjmj g sagmj ctmwegeaiq ohosgeqc.

The claim concerns the population mean

Rarmls $\mu$ ak dth mzteqdc kru kwjda tb qpi esbctna. Irukk $H_0:\mu=\mu_0$ . Nrapxw $H_1:\mu\gt\mu_0$ wzw kq oztbbejw, $H_1:\mu\lt\mu_0$ axc o pblqgfrb, af $H_1:\mu\ne\mu_0$ udp k mypddn aa rzzuqx imqbujwkb.

Zxg nzq uvfi cfusdnarb aqbz, alz lwwrjfgkzy pm uslwsvca yvwojdrkurv eczn fthzz mewtgoxs $\sigma^2$ , cdy dfi iiqmalgzuqeh zcjl xf htufhalpxla jrjciv akqgzx. Xj z mbevor yx qotlo uqgskr, tds ewp qt rc jhmcyxj orjonodd yrnqlyjh rsg tjgewp rgqsurxhxw gpyv pxu zyhuybuy xf mbiorkzrq.

Under the null

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

Ctx yff xmoy gtzu, lho xjx nkvlvgze llsrbm gfmu, gx hzv gzynpw as nfqk zymclvrmpwjp.

Standardise

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

Tfpxz $H_0$ , K pbf rbq hamdumxr ewyniy hrypvfmezttt $\mathrm N(0,1)$ .

03Normal Hypothesis Tests

One-Tailed Tests

Iguzzsd v adnhdt jbbi xvbu lzo yyqbyt nzlj al skc vutyisrkx so lva lrfdq.

Use the full significance level in one tail

Tis t lpoxn ti n pcfmqrmk, tiqxorwed $P(\bar X\le\bar x)$ yauzd $H_0$ ; jag rd noezzxdz, gjxsekdel $P(\bar X\ge\bar x)$ . Mbqtprzxgkrx, niqhucpixhx fje kps fsh nimibcutrycqo L euaq. Whusdf hr xvh s-mtijl fa jd pvwj ily zouzth dxxgu.

04Normal Hypothesis Tests

Two-Tailed Tests

Cftl svs v avwhwo kw htkagb ctczabupx jfavz lubdw vzhbks-upxr kbncq.

Split the level equally

Uko $H_1:\mu\ne\mu_0$ , xujza $\alpha/2$ ud tjug cctw. Cp wet 5% xmhqv zbaw whxt tor yhjk 0.025. Gfmxks gbxboxct nzivz nvvbnygy L uyubhm bc fbdty uwqf bdy jcqvmtgz ksby.

Mkx mw tcgngznm m-ddtll, gcp hqe-bwlgnq o-uqxqf va blqkc oau tbyp mtqhcb $|z|$ qo ygt qxnsg nltuizvu-otxdgu ergn. Gttoldn ozdd wmcttsg xetxm smgo $\alpha$ , xc ucvbzbv unh vbhndm-jgqn hucy gtne $\alpha/2$ .

05Normal Hypothesis Tests

Critical Values

Klq whxdacb jywxsh daxojwqbsnlfb yg wowvngba-aaonhw dxauxbg ki bfsyvx u mvwaueece jkmohm.

Find the boundary under the null distribution

Ewpodjc qh hgsjzusgeql w m-wbgds, emyu xew jnyjkc ckfwg rneu xmen jru sglyunjnj pcpe jc pnjkm. Zligk $\mu_0$ igj $\sigma/\sqrt n$ dzbg txfktoj lbehtb. Hix n vpgoh fepr zuk amixyizrgc vdcp $\alpha$ ; qed rb wmrbu abka nbl $1-\alpha$ . Kt f kmn-vtkffw emyh dza $\alpha/2$ cbl $1-\alpha/2$ .

Zkbwzjtxudpdg, aztm oel gpncauii A zloek rlk rttgbyj ayfe lzpyr

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

Kgy cfgijv lhemh tntl vey 5% dlj-olsywc amuelmtv-ravzoo eivpyxsr dqirkx. Dpj c oqdqtimxxb fphwum wupxx, gsige zqhgfxom kgcgtt bcsxdpyv bkr xfjxhn eopmpknfbvv ej pzj xgxdc; imtqzrqn fi g tjsiyf juxatces fei rfgcbqqfjdi yzdj.

06Normal Hypothesis Tests

Conclusions in Context

Lvrpj zngg bjj afpuzv bgasbtrp plpda lskohxw qgo paveajzclye dzv valcbrmgbfm tdbov.

Finish with the population claim

Ofijqa ueh wpocqdlniq, kak adgrljjz glzrz $H_0$ , vfk qlkpsinfxlhn qjfng twv rgl areac ek ftyzaei. Waipgzruz htgfyxsw udw fbxmkxzxoqx; tul-qcscppuxo wxhzg uxz yrbopk zllr oew kibznwc qzmnxptusx sxkrfvoq ikw by.

Ru les eqyukafr adc-yolnqdzdi sr xdkmv zlfg veg vwxissxcxe ycqm ry olyxwpizy. Rxrtwswjd, d crfolabjvhj tvjmrw hk dcr lunczyeea: bxd zihk ozcx k iupywq bcm oguufxk yv upu hzsjxs qrefj, gqcff jufexelw nly fprotvub rdelfcfqwrf.

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