FWSx_ C[x. 5\ ~ x݃'5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ (e?HJgJsQJylh/Rm;O`GPkQ_P̟!f{>;+i!I'.Y 0$,hI. QHEZ?[(KdRब b@F@0 9#hi dZl.Tm!))iG)EʖWʏK+rK!y=ֿ3+`L~O8J&̞7BNqQ"@\[kob>K uHd<o&;8zNj3-4} /5ȕm c7^21GҐѰ /|G9^廌{:`2wx͸ .VEK'Zvuôev7r7咒uد.Q9%.Fd? [ Verdana!DFUc\%/G5q1n"`m 5L-6ix5/kurw5ggqlZ 1 =%ޗӹON4+RJe"YOdU6e%\4k_R7YeΆG?|Ozp3 E8B5b.o6}ʓ.nWyTce3mٴMP5hYzև]'p3m[dvUx 4X`$ʘ$P`k~Z`%Y r/f^T3 & 5hej.3%# L Q4=ʥڮm#FYa ,N6<9'+M@mC'R$^^`ci3eZku̘[*!&YgGz{ DeFٗڿkTE"-*CJM5L]8 @ D+D7lHv(7e5h* 6u.U“A&+BM n#Nl; ^rgv̾CʤrR$_iP5E.)EbP9\m[mg l5ffCUYڽP&Ta.C"kT@ dZ{ޛW_n 'Em&/Řdn lkP'e"7TNlB*]?+ ҐZt܋Z Rb@ @Q^Xpj$\o`5i.+̳V cfo5fdvųk)Jxҍ<ڟ*i 0( э]`.T4&9~\١]jGkvf鏯B%rʹk}ijaFn #&Kle\A)EЀ5I.GlDnܶRK0lDnܶ7Q5kh8 Յ 1ɋ1V5k3-~͇"QPk3bـuR.iCց۔NZ1ޮLX85hvV8"Z1ʤ8@5jrxx؍sbS Fq͇.&J]ʹ5jvV9-'3s8$DS~`5feV)yP R)6^(bc]cR\$m3:l )@A9fSvjϥ#zg)@`MkI,O5*ZBȽY%`@0:bS;vin6̈́m5gZr.Zn&6D`RiM5iiv̅gz6 r.l7fH /e2]ݪ 5DWP&כobk@ Bɴ~Q ̗^3Ǩ@Z&Y@3R ф6JRcVBKDfg6[Y[Ppܗ5f&}U/]{ FOqSZw(b  f1˘)t[z5hx^ ~Sf!e.%ݬSeAЎyeYkF 1n'ـ.s%)-P]_Uڤ{Hl,Pm3@9L&!1MH<%Sn>[[yہ5byQv-^c lKl5l5b{yXV,^5ef[/7jآᰶKMg];k[ +`aQr-WҬd5I,*b i%5r˴XmSPɀJ*6LA8F̍`32bx8't4PՌ5fvX2/]: N3bl/lJ⚽Ҫڴ`u 5ff1\Yp fR['2e*Ixnkj.d 2IR%` d%gfqf[!eZI[dWenv\۳n\6Iؤ:h敜ymf 5a~wk ۪~u @1Y Q2&%@ l R@+* 5iy]Cϐ]-n%kyī6 -.012345679:<=DLXZacefgilnorstuv g2 2(

H  :    = 30

 hWb 2(

0

 wg;_ 2(

Test statistic:

 g2 2(

H  :    < 30

 hWb 2(

1

 w_v`2(

Z   =

 h W^ 2(

x

 XFx @@ P, `C hn A >rH@I,Tf *0BPA‡Q@ n g 2(

= 0.05

 g5 2(

Reject H   if

 hWb 2(

0

 hW^ 2(

x

-`  @Tk@H])PX D@d $g)#XyeW$7P F@P6ֆFXG$X~`e  D?NP Ms!8m!J!IE6ܤmGGZnnvܣͨ;nn4fԄ} #8-]?W"$U&:lKd5fw s7p wn`7pq5lw w@rwn70Kr @n5ly0 n8(n9(,`5j@xWp r`8h5d@y\ 85j@x u@n`7P u@p` sp,@8X q@n5H@J 8V5hw480sn70q@n5lws -@n9 <`8(U5ix(^{[s, 95lw(u@n 7P n70  rFjpP(À5jx(n7(u`7p `8pLp "@n 7P5l@wP+9h`5j@xV@rX p 5d@yTun 9lʠ70 n`7p[5jwP 8p spn 7p @8P q@p @n`5standard 07_63 >CEHIKLNOPRSTUVX!XgPd %%AFRP ss Ms!pS6ҢnR6M7;nQf{7TjB>ԋT럁+݌$`'``59EpƝ W= checkMovie7%1`  zpB-So2B= w 2(

Reject H   since -1.743 < -1.645

 hWb 2(

0

 @'KPg@'N n2@'KP@@'N  M Z@ a8 w @ Mf j@U@@#D `@#T`@#^X!쀏<@J<@QU@<@Sq&<@Y=60<@]@'7L@'GAR@? checkMovie7 !g2 2(

H  :    = 30

 h"Wb 2(

0

 w#g;_ 2(

Test statistic:

 $g2 2(

H  :    < 30

 h%Wb 2(

1

 w&_v`2(

Z   =

 h'W^ 2(

x

 n(g 2(

= 0.05

 )g5 2(

Reject H   if

 h*Wb 2(

0

 h+W^ 2(

x

W,= checkMovie6% ^-!@'KPg"@'N n#2$@'KP@%@'N  &M 'Z@ a8 w @ Mf(j@U@)@#D `*@#T`+@#^X!쀏<@J<@QU@<@Sq&<@Y=60<@],@?-checkMovie66.x. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ (e?HJgJsQJylh/Rm;O`GPkQ_P̟!f{>;+i!I'.Y 0$,hI. QHEZ?[(KdRब b@F@0 9#hi dZl.Tm!))iG)EʖWʏK+rK!y=ֿ3+`L~O8J&̞7BNqQ"@\[kob>K uHd<o&;8zNj3-4} /5ȕm c7^21GҐѰ /|G9^廌{:`2wx͸ .VEK'Zv /g2 2(

H  :    = 30

 h0Wb 2(

0

 w1g;_ 2(

Test statistic:

 2g2 2(

H  :    < 30

 h3Wb 2(

1

 w4_v`2(

Z   =

 h5W^ 2(

x

 n6g 2(

= 0.05

 7g5 2(

Reject H   if

 h8Wb 2(

0

 h9W^ 2(

x

W:= checkMovie5% ;./@'KPg0@'N n122@'KP@3@'N  4M 5Z@ a8 w @ Mf6j@U@7@#D `8@#T`9@#^X!쀊:@?;checkMovie50<x. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ (e?HJgJsQJylh/Rm;O`GPkQ_P̟!f{>;+i!I'.Y)+bؐ:CDBv30 p.bBL K#d`/KxR3z֏6:.Tm!))iG)EʖWʏK+rK!y=ֿ3+`L~O8J&̞7BNqR(ߙOJ%ȅ:|2OW;8zNj3-4} /5ȕm"@\[h(r41>Uem{9@7F^5wuôev7 rDۋB\!a"Ėmpd =g2 2(

H  :    = 30

 h>Wb 2(

0

 w?g;_ 2(

Test statistic:

 @g2 2(

H  :    < 30

 hAWb 2(

1

 wB_v`2(

Z   =

 hCW^ 2(

x

 nDg 2(

= 0.05

WE= checkMovie4% F<=@'KPg>@'N n?2@@'KP@A@'N  BM CZ@ a8 w @ MfDj@T|ـE@?FcheckMovie4Gx. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ (e?HJgJsQJylh/Rm;O`GPkQ_P̟!f{>;+i!I'.Y)+bؐ:CDBv30 . R^֐B@16$ 0E^.Tm!))iG)EʖWʏK+rK!y=ֿ3+`L~O8J&̞7BNp.[ǽip0j]\Ifg@+HXw"%n- Hg2 2(

H  :    = 30

 hIWb 2(

0

 wJg;_ 2(

Test statistic:

 Kg2 2(

H  :    < 30

 hLWb 2(

1

 wM_v`2(

Z   =

 hNW^ 2(

x

WO= checkMovie3% PGH@'KPgI@'N nJ2K@'KP@L@'N  MM NZ@ a8 w O@?PcheckMovie3Qx. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ "[e?HJgJsQJylh/Rm;O`iLlzX4yf{>;+i!I' Rg2 2(

H  :    = 30

 hSWb 2(

0

 Tg2 2(

H  :    < 30

 hUWb 2(

1

WV= checkMovie2% `WQR@'KPgS@'N nT@'KP@U@'N V@?WcheckMovie2Xx. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ Yg2 2(

H  :    = 30

 hZWb 2(

0

W[= checkMovie1% >\XY@'KPgZ@'N n[@?\checkMovie1?fulltargets?initframeiadragdragoriginx_xoriginy_ytarg_root.targetonPressthisstartDragonReleasestopDrag_droptarget_rootfulltargetsnextFrametargetNotargetNoHGNONO GO    RO   R N NIL NIg  NNO  NNONPO  R*  NO  NONI ROPC?initframe=]u] \d ssλ2=Z|sZcpZ;Y+JtThe drying time for the paint in a section of a manufacturing process is known to be 30 minutes, with a standard deviation of 2 minutes.  The process engineer hears of a new brand of paint, which is advertised to have a reduced drying time.  He paints a sample of 15 items using the new paint, and measures the drying times.  Using his calculator, he finds the mean of his sample to be 29.1 minutes and the sample standard deviation to be 2.6 minutes.  Using a 5% level of significance, what should the engineer decide?

For each of the steps in the solution, drag the correct working into the empty box.  

`%p? aVerdana zbg9 2a(

Explanation

b@X!o xco2(

Null Hypothesis

c#A &dh51[NT` ed@eȃ@.$f`3@`59DpƝ gf@&g]6target3$h`3@`59DpƝ biZcĵlXTT|Y_sY vc 7]5 \nEaVٓ_e?HJgJsQJylh. jg2 2(

H  :    = 30

 hkWb 2(

0

&lh5!8dJ Lm?hijȠk@?lfm @&m@#_(@drag3n`3@`59DpƝ >jXSt; 4*ZA_*>U,,ӻ1Zt̮OgdA7g"İώDHRl. og2 2(

H  :    < 30

 hpWb 2(

0

q`3@`59DpƝ >lXTT|Y_sY vc 7]5 \nEaVٓ_e?HJgJsQJylh. rg> 2(

H  :    < 30

 hsWb 2(

0

_t?noȠp@?qrȠs@&t@#2Q(@drag1uZveA9%i9 vg4 2(

H  : X  < 30

 hwWb 2(

0

Sx?husvȠw@?lfm @&x@#(@drag4 ~yg5 2(

H  : X  = 30

 hzWb 2(

0

L{?huyȠz@?lfm @&{@#FJdrag2@ |w +"2(

Hypotheses are about population parameters (Greek letters).  The null hypothesis has the equal sign.

 |@X!%}Xt@Pd Bh(&@%~Xt@Pd ssBh(&@$]l70P5Mv`t!$`" @ JhĠ<&BdOF}@ 8~@ 8}@ 82@  P@  ?fulltargets?initframe= oU2(

Alternative Hypothesis

 @ȃ@2񗀿6gf6target20h,@d %( JP4gy*Pb0h,@d ss%( JP4gy*Pb%o`5(Dp B @P @P @P @P/checkMovie1ClipcheckMovie1_root attachMovieR &h5!8dJ g2 2(

H  :    < 30

 hWb 2(

1

K?fm iȠ@?l@6@#FJdrag2 g2 2(

H  :    > 30

 hWb 2(

1

K?fm iȠ@?l@6@#_(@drag1ZveA9%i9 g5 2(

H  : X  < 30

 hWb 2(

1

K?fm Ƞ@?l@6@#2Q(@drag4oaHP P%]VOҜ&|Ҟ[4Z i|T$NǓo=k2j==?8ݟ>;+i!I' g2 2(

H  :    = 30

 hWb 2(

1

K?fm Ƞ@?l@6@#(@drag3@  w J"2(

Alternative hypothesis is what the experimenter wants to establish: that the new paint dries quicker.

 @X! P@  ?fulltargets?initframe= vg7 2(

Test statistic

 @ȃpN"zoN"aB @P @P @P @P/checkMovie2ClipcheckMovie2_root attachMovieR &h5pyY!؇V`0p(<X{:ZZt̮OgdA7g3ܙq[Hi M=St; 4*ZA_*>U,,Ժ%`lT2Haȁ@HN[ ff[=z[ZBL K#d`/KxR3z֏6:[\ qhK -ip0j]Xc?Vϋ"%Hrl7 w_v`2(

Z   =

 hW^ 2(

x

&h5!8dJ h?_@Zk'` فh DJ@?fИ@6<drag2a(- *WCiݘ{MG~:fWMCg2~L~O8J&̞)BS:SҏSfAsV‚Z.GZ\;LWnp+pg@ȱ _v`2(

t      =

 jW{ 2(

n-1

P s?_@Akz` Ahʘ J@?fИ@6 drag1`0p_tݔ!))iG)EʖWʏK+rK!.'#5LlzOv~3=ɟVٓ\յ|Wc9n| Ihl#h.D$ 0$,hI. QHE%,X&Ā,ar F&ٙ w_v`2(

Z   =

 hW^ 2(

x

]?_@-(5  ل@?fИ@6 b<drag3a(- *WCiݘ{MG~:fWMCg2~L~O8J&̞)BS:SҏSfAsV‚Z*%`lT2Haȁ@HN[ ffb^֐B@16$ 0E^3FGRYM2:`2wėcn3  _v`2(

t      =

 jW{ 2(

n-1

h?_@Akz` Ah J@?fИ@6&d<drag4@ yNT%OKiWnbmI0%ׁ<`)kGaEIz[ZBL K#As!"mm!;lKM6*M\a  w J"2(

Population standard deviation is known, so use Z test.  Standard error of the mean is  

 @X! P Kr@   &?fulltargets?initframe= ~o2(

Level of significance

 @ȃ@/ї6g?0!@target1B @P @P @P @P/checkMovie3ClipcheckMovie3_root attachMovieR -`3@`58pF}t$W yQ(L;"<$瞧Ӛ6.Ŗ d#zN .C-ch5OJ%FuHd8*QX9L{@&!`YGqF^9 ng 2(

= 0.05

B?@?lfm @6dPdrag1 ng 2(

= 0.01

B?@?lfm @6Pdrag3RVLMgre;JH]`Y wA%l|ۨ-!@OpZkRUE+׀N<>z-kBm"ͬ H6qOn8'ȉ( ng 2(

= 0.05

B?@?lfm @6 bPdrag4 ng 2(

= 0.01

B?@?lfm @6 drag2@  w JX2(

5% significance, so the probability of type I error is 0.05.

 @X! P@  ?fulltargets?initframe= ug8_ 2(

Decision rule

 @QtaЍ̜%$bpB @P @P @P @P/checkMovie4ClipcheckMovie4_root attachMovieR 2`>!5P}8C>K(UBr&X/H%d vYE[  Lg=W"2(

Reject H   if

Z   < -1.96 or

Z    > 1.96

 hWb 2(

0

 hW^ 2(

x

 hW^ 2(

x

b?Ph>`@@?lȠ@@6 b+drag2 PoE/2(

Reject H   if

t   < -2.1448 or

t    > 2.1448

 hWb 2(

0

 iW{ 2(

14

 iW{ 2(

14

Y?Ƞ@@;;@?@6`+drag4Y46Ut%do g5 2(

Reject H   if

 hWb 2(

0

 hW^ 2(

x

q?Ƞ@@#F2@#I@@#'ʜ#@?l@6b+drag1 g9X2(

Reject H   if

t   < -1.7613

 hWb 2(

0

 iW{ 2(

14

^?Ƞ@@#2<@#GAT@#L@?l@6"drag3@  w J 2(

Critical value for one tail Z test.

 @X!  P @  "?fulltargets?initframe= tg< 2(

Computations

 @ ȭp#@6gQ,z$ target2B @P @P @P @P/checkMovie5ClipcheckMovie5_root attachMovieR  5`X%mmu:`2w喻=Hf@XEe <` hW{X2(

14

P FXWI$X~`e  D? {? .`0 @,ȉ@?lȠ@@6p drag1 hW{X2(

14

$X~`e  D? y?.`3' @ Rȉ@?lȠ@@6_< drag4`X@%N{f@$Xw@e  g$ c?ɏЉ@?lȠ@@!"HW n?.`@ Rȉ@?YИ@6( b< drag260B< drag3@ yad&7YoTT|Y_sY vc 7]5 \ng3<ⶑ*2z LNvJ>iO--r-*K`HY&&}]xִ~Q*4%`lT2Haȁ@HN[ ff  w JX2(

Substitute values   X = 29.1,   = 30,

 = 2, n = 15

 @X!  P @   !(0?fulltargets?initframe= pg%' 2(

Decision

@ ȃp#6gzn$ target1B @P @P @P @P/checkMovie6ClipcheckMovie6_root attachMovieR   pg2 2(

Reject H

 hWb 2(

0

D?fИ2#@?@6]< drag1 g2X2(

Do not

reject H

 hWb 2(

0

C?fИȠ@?@6< drag2@  w JX2(

Obtained Z value is in the critical region.

 @X!  P @  ?fulltargets?initframe= rg/W 2(

Conclusion

 @̀!l`aпB @P @P @P @P/checkMovie7ClipcheckMovie7_root attachMovieR  o;/2(

The engineer should use the new brand of paint

3?@?@6(+drag1 o/2(

The engineer should continue using the old brand of paint

3?x@?l@6"+drag2@  w JX2(

Rejecting the null hypothesis means we accept the alternative.

 @X!!Xe@Pd 1@hF!Xe@Pd ss1@hFOF@ 8@ 8@ 82@  @  ?fulltargets? initframe=@