FWSƕx_ Cx. 5\ ~ x݃'5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ 'A}!))iG)EʖWʏK+rK!ΛQ_P̟!dXq[Hi M0\)BS:SҏSfA}- *WCiݘ{MG~:fWMCg:7g3ܙq[Hi M=rfȐ!t؏tdB{>'FɎ$瞧Ӛl;Lt 2_H1Mr%[`)damQ4l(#h>r&qcδv5q .W;qhK,82\¾0.S}@ip0j]\zQbـuR.iCց۔NZ1ޮLX85hvV8"Z1ʤ8@5jrxx؍sbS Fq͇.&J]ʹ5jvV9-'3s8$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

Chn A >rH@I,Tf *0BPA‡Q@ n g 2(

= 0.05

 _v`2(

t      =

 j W{ 2(

n-1

 XFx @@ PP, ` o2(

Reject H   if t   < -1.7613

 hWb 2(

0

 iW{ 2(

14

XEe <` hW{X2(

14

#XyeW$7P FXWIP6ֆFXG$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(

Do not reject H   since -1.341 > -1.7613

 h Wb 2(

0

s!@'KPg@'N n2@'KP@@'N  @ Mf j@U@ !K@ W 6 1Zu@#C00@#Spp@#\(`B;J+RP@n;@',L @'IR@?!checkMovie7 "x. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ 'A}!))iG)EʖWʏK+rK!ΛQ_P̟!dXq[Hi M0\)BS:SҏSfA}- *WCiݘ{MG~:fWMCg:7g3ܙq[Hi M=rfȐ!t؏tdB{>'FɎ$瞧Ӛl;Lt 2_H1@Frf5?Xc#T}) ;8~MH9K\Z\;LWnp+W"ؖEpd rDۋG%o֗!۸ɩ́rD^0 #g2 2(

H  :    = 30

 h$Wb 2(

0

 w%g;_ 2(

Test statistic:

 &g2 2(

H  :    < 30

 h'Wb 2(

1

 n(g 2(

= 0.05

 )_v`2(

t      =

 j*W{ 2(

n-1

 +o2(

Reject H   if t   < -1.7613

 h,Wb 2(

0

 i-W{ 2(

14

 h.W{X2(

14

W/= checkMovie6% Q0"#@'KPg$@'N n%2&@'KP@ '@'N  @ Mf (j@U@ )!K@ *W 6 1Zu+@#C00,@#Spp-@#\(`B;.J+RP@n;/@?0checkMovie61x. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ 'A}!))iG)EʖWʏK+rK!ΛQ_P̟!dXq[Hi M0\)BS:SҏSfA}- *WCiݘ{MG~:fWMCg:7g3ܙq[Hi M=rfԀ~F6R6UweH9K\ټ2$qEm#]$g^Cc39殛8L rS\Vqcδv5q .W0E ,zӷ 2g2 2(

H  :    = 30

 h3Wb 2(

0

 w4g;_ 2(

Test statistic:

 5g2 2(

H  :    < 30

 h6Wb 2(

1

 n7g 2(

= 0.05

 8_v`2(

t      =

 j9W{ 2(

n-1

 :o2(

Reject H   if t   < -1.7613

 h;Wb 2(

0

 i<W{ 2(

14

W== checkMovie5% >12@'KPg3@'N n425@'KP@6@'N  @ Mf 7j@U@ 8!K@ 9W 6 1Zu:@#C00;@#Spp<@#\(`=@?>checkMovie5?x. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ 'A}!))iG)EʖWʏK+rK!ΛQ_P̟!dXq[Hi M0\)BS:SҏSfA}- *WCiݘ{MG~:fWMCg:7g3ܙq[Hi M=rFGҺIv2!@ν "<$瞧Ӛl;Lt 2_H1@F$HK{X: NF21GҐѰ 4s$ ѓ~k5ƹqcδv5q .W0E ,zӷ @g2 2(

H  :    = 30

 hAWb 2(

0

 wBg;_ 2(

Test statistic:

 Cg2 2(

H  :    < 30

 hDWb 2(

1

 nEg 2(

= 0.05

 F_v`2(

t      =

 jGW{ 2(

n-1

WH= checkMovie4% I?@@'KPgA@'N nB2C@'KP@D@'N  @ Mf Ej@T|ـ F!K@ GW 6 1ZuH@?IcheckMovie4|Jx. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ "[e?HJgJsQJylh/Rm;O`iLlzX4yf{>;+i!I'ؓ_e?HJgJsQJylh/Rm;zet6z{=s'~q?%|w?'V%ZBfL.,c9֗!۸ȱr%q\Q:v Kg2 2(

H  :    = 30

 hLWb 2(

0

 wMg;_ 2(

Test statistic:

 Ng2 2(

H  :    < 30

 hOWb 2(

1

 P_v`2(

t      =

 jQW{ 2(

n-1

WR= checkMovie3% SJK@'KPgL@'N nM2N@'KP@ O@'N  P!K@ QW 6 1ZuR@?ScheckMovie3Tx. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ "[e?HJgJsQJylh/Rm;O`iLlzX4yf{>;+i!I' Ug2 2(

H  :    = 30

 hVWb 2(

0

 Wg2 2(

H  :    < 30

 hXWb 2(

1

WY= checkMovie2% `ZTU@'KPgV@'N nW@'KP@X@'N Y@?ZcheckMovie2[x. 5\ ~ x݅'τ5H ~ x݅'τʐ[Ҝ&|Ҟ[4: |T$NǓo=k2j== q?ώDHRlh/ \g2 2(

H  :    = 30

 h]Wb 2(

0

W^= checkMovie1% >_[\@'KPg]@'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+JRead this question very carefully to see the difference from exercise 1.

The drying time for the paint in a section of a manufacturing process is known to be 30 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.

d%H zeg9 2c(

Explanation

e@X!o xfo2(

Null Hypothesis

f#A &gh51[NT` hg@hȃ@.$i`3@`59DpƝ ji@&j]6target3$k`3@`59DpƝ dlz@ԓ&7oTT|Y_sY vc 7]5 \ng3<ⶑ*2z LNvJ>iO-- mg2 2(

H  :    = 30

 hnWb 2(

0

&oh5!8dJ Pp?klz>mȠn@?ofm @&p@#_(@drag3q`3@`59DpƝ >jXSt; 4*ZA_*>U,,ӻ1Zt̮OgdA7g"İώDHRl. rg2 2(

H  :    < 30

 hsWb 2(

0

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

H  :    < 30

 hvWb 2(

0

_w?qrȠs@?tuȠv@&w@#2Q(@drag1xZveA9%i9 yg4 2(

H  : X  < 30

 hzWb 2(

0

S{?kxsyȠz@?ofm @&{@#(@drag4 ~|g5 2(

H  : X  = 30

 h}Wb 2(

0

L~?kx|Ƞ}@?ofm @&~@#FJdrag2@ w +"2(

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

 @X!%Xt@Pd Bh(&@%Xt@Pd ssBh(&@$]l70P5Mq-^G Ї*$`" @ JhĠ<&BdOF@ 8@ 8@ 82@  P@  ?fulltargets?initframe= oU2(

Alternative Hypothesis

 @ȃ@2񗀿6jf6target20h,@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

O?fm lz>Ƞ@?o@6@#FJdrag2 g2 2(

H  :    > 30

 hWb 2(

1

O?fm lz>Ƞ@?o@6@#_(@drag1ZveA9%i9 g5 2(

H  : X  < 30

 hWb 2(

1

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

H  :    = 30

 hWb 2(

1

K?fm Ƞ@?o@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"6jzoN"atarget1B @P @P @P @P/checkMovie2ClipcheckMovie2_root attachMovieR &h5pyY!؇V`0p(8 j=k2j==?8ݟrgsym"U)6dOҜ&|Ҟ[4Z i|T$ORx{յw$6LM &'E#=hl#h,%`lT2Haȁ@HN[ ff bD"֗!۸%l)T.Vp w_v`2(

Z   =

 hW^ 2(

x

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

t      =

 jW{ 2(

n-1

s?_@Akz` Ah ʘJ@?fИ@6 drag1`0p_tݔ!))iG)EʖWʏK+rK!.'#5LlzOv~3=ɟVٓ\յ|Wc9n| DK9+bؐ:CDBv30 ($$6LM &'E#= w_v`2(

Z   =

 hW^ 2(

x

]?_@-(5  ل@?fИ@6  b<drag3a(- *WCiݘ{MG~:fWMCg2~L~O8J&̞)BS:SҏSfApXCץ$6LM &'E#= p6QԖ z b@F@0 9#hi dZltAn- bi#. \C+qKg3 IwX _v`2(

t      =

 jW{ 2(

n-1

h?_@Akz` AhJ@?fИ@6'd<drag4@ 2yNT%OKiWnΛ   w J"2(

Population standard deviation is not known, so use t test.  Standard error of the mean is  

 @X! PKr {[@    '?fulltargets?initframe= ~o2(

Level of significance

 @ȃ@/ї?0!@B @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?@?ofm @6dPdrag1 ng 2(

= 0.01

B?@?ofm @6Pdrag3RVMaSZkRUE+׀N<>z-kBm"ͬ H6qOn8'ȉ( sN 6}˕# +!ud[ޱtX nC BM= ng 2(

= 0.05

B?@?ofm @6 bPdrag4 ng 2(

= 0.01

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

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

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

Decision rule

 @Qtaп6j̜%$bptarget3B @P @P @P @P/checkMovie4ClipcheckMovie4_root attachMovieR 2`>!5P}8C>K(UBr&X/H%d vYj  Lg=W"2(

Reject H   if

Z   < -1.96 or

Z    > 1.96

 hWb 2(

0

 hW^ 2(

x

 hW^ 2(

x

b?Ph>`@@?oȠ@@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

-`  @Tk@H])P q?Ƞ@@#F2@#I@@#'ʜ#@?o@6b+drag1 g9X2(

Reject H   if

t   < -1.7613

 hWb 2(

0

 iW{ 2(

14

^?Ƞ@@#2<@#GAT@#L@?o@6"drag3@  w JX2(

Critical value for one tail t test with 14 degrees of freedom.

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

Computations

 @ ȭp#@6jQ,z$ target1B @P @P @P @P/checkMovie5ClipcheckMovie5_root attachMovieR  5`X%mmu:`2wR.Fd Ykb` hW{X2(

14

{? .`0 @,ȉ@?YИ@6p drag1 hW{X2(

14

P F@$X~`e  D? y?.`3' @ Rȉ@?oȠ@@6_< drag4`(0%NSf@X D@d $g)$Xw@e  g$ c?( @O b@?oȠ@@!"HW $X~`e  D? n?.`@ Rȉ@?YИ@6( b< drag260B< drag3@  l w JX2(

Substitute values   X = 29.1,   = 30,

s = 2.6, n = 15

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

Decision

@ ȃp#6jzn$ target2B @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 t value is not 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@?o@6"+drag2@  w J"2(

Not enough evidence from this experiment to change the existing practice...

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