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Unformatted text preview: Fall 2009 STATISTICS 401C
Examination 2 (100 points) Name NOTE: Please show all work to obtain full credit. 1. Trace metals in drinking water affect the flavor and an unusually high concentration can pose a
health hazard. Ten pairs of data were taken measuring zinc concentration in bottom water and
surface water from various locations of a stream. Do the data suggest that the true average
concentration of zinc in the bottom water exceeds that of surface water? The data are given
below aggregated: Location Bottom Surface Difference cl 1 .430 .415 .015
2 .266 .238 .028
3 .567 .390 .177
4 .531 .610 .079
5 .707 .605 .102
6 .716 .663 .053
7 .651 .632 .019
8 .589 .523 .066
9 .469 .411 .058
10 .723 .612 .111
Ed = .55
2ch = .072194 Let W = abattom — usurface be the mean difference in zinc concentration of the two populations and
assume that the differences dz have an approximately Normal distribution. (a) (10) Compute a t—statistic to test the research hypothesis that the mean zinc concentration
in the bottom water exceeds that of surface water. State the null and alternative
hypothesis in terms of ,ud. Compute an approximate p—value and make the decision using oz = 0.05.
go m. Illa/«45>o Ho: M
A; ‘__ 472/434 “Mgr/(0 __ mot/{A
7 (b) (10) Construct a 90% conﬁdence interval for ad. Use this interval to test the hypothesis in
part (a), explaining how you arrived at your conclusion. What is the (Jr—level of this test? 9 m ~——~ . A
10” C'L ' d If ewes '76%
‘0'” v be: :1: T) (momajarwva)
99mg 0 a m Ith w at». xtth
We veg/“eat” )‘ﬂL WM/wféﬂ's mug" 01:05 2. Many people purchase sport utility vehicles (SUV’s) because they think they are sturdier and
hence'saier than regular cars. However, data have indicated that the costs of repairs for SUVs are
higher than for midsize cars when both vehicles are in an accident. To verify this observation, a
random sample of 9 new SUVs and 8 midsize cars are tested for front impact resistance. The
amounts of damage (in hundreds of dollars) to the vehicles when crashed at 20 mph head—on into
a stationary barrier are recorded below: Vehicle Damage (in $100’s) g
SUV 14.25 24.47 18.15 15.17 27.48 16.42 29.56 12.33 21.54 9 19.93 37.6349
Midsize 11.95 15.42 13.27 9.87 18.12 10.36 12.65 21.25 8 14.11 15.4741 Let #1 and 11.2 represent mean cost of repair for the two types of vehicles, respectively. ‘ (a) (10) Just by looking at the sample statistics, can you tell if the assumption of equal
population variances is plausible or not? Explain. Perform a statistical test using
a =' .05 to verify this assumption. State the null and alternative hypotheses and the
rejection region clearly. 2
 a r. _l.. .
new “ Et“* .5: “‘
'07—3’3772
rings/7 g. 4,620 ’17 Kg, F <1?"
mrF7%% (b) (10) State and test the appropriate hypotheses to determine if the mean repair cost for
SUVS was higher. Use o4 = .05 to make your decision. Mot/Ac’ﬂa <0 m. «liar/«.790 A“ : Qua) Afar(wro/sfz H37~éavj+7W<§o4¢7ﬂ£g7wg
k ’VLIi 91,—2— l5" 2:; AP —= 5122543 A], {Ea.157, .2537
“QQ. ‘6 7 6,05, ,5 Ate. 671.75? ‘ ' ' *7
%m.r fa 25 m ML RR. and its/.113 4190(205'
(c) (10) Construct a 90% conﬁdence interval for m — ,ug. Test the research hypothesis that the
repair cost for SUVs is larger by $1000 on tl_ie_average, using this conﬁdence interval. 702’ CI Hy #702: %,—?,iﬁ%)yp'/‘plm it“). gas/452»! . (Wagx442) .2: #753 22537 5.82. :l: 4.45 (£37 / 0.27) 4431/41“/AJ<IO m. ,ﬂl— 2>lO ! r V
<s>maa 10 73 (m %s ?7’I’/?~/um€,) we ‘éw/ 42’ 74m
3. A research engineer at an electric utility is investigating the use of a Windmill to generate
electricity. He has collected data on the Wind velocity and the corresponding DC output from his
Windmill: Wind Velocity (mph) DC Output (m) (y)
5.0 1.6
6.0 1.8
3.4 1.1
2.7 0.5
10.0 2.2
9.6 2.3
8.2 2.2
3.0 0.7
5.5 1.6
7.0 1.9
8.8 2.1
9.1 2.3
2.7 0.7
4.1 1.2 Some summary statistics are: n: 14 23331 =85.1 Zyi 222.2
Zr? = 611.85 2y? = 40.52 Exiyi = 156.43 (a) (8) Fit a linear regression y = ,80 + ﬂl a; + e of DC output on the Wind velocity using the above data. State your least squares grediction equation. 82.51 :2. 22% r. /h :2 AME”: "€95.1f/l’1 :: 44154357
337 a :Wv Ex?" : IS’éJ/Ds ~ @§.I)sz.,z)/H_ zil~$€ﬂ§7l
A?" ____ so? km: atrial/(#54357 = 0.2171,
A ~ A " z 
(1% a: U —' , a? :2 _Q2273.I)(£7‘%L : &.2&7¢él
Preaér‘cﬁm 9 .2 0.20%; .+ 0.227921% (b) (7) According to your ﬁtted model, What is the average increase in DC output associated
with a 5 mph increase in wind speed? Must show work. AWfé Warfare 4’”, LDC am‘lawe“ 4}» I 451mg.th
407‘ s ~.= 52272] WM MW [IA/W498 L'h/ 35C Emwu AMMW 2M Mhag S/aeec/ : 0427.21 )(5 (c) (8) Fill in the analysis of variance table for the regression below. What is the estimate of
the error variance 0?? Compute the F—statistic and use the F table to test H o : ,61 = 0
vs. Ha : £31 aé 0. Use a: = .05 and show work. Source d.f. SS MS F
Regression ( "fjﬁﬂQ— Lfvgé’le? I 34/, 5
Error UL 0, #3251}, 0. 03é2& Total 13 15,3 ,7 I 2‘12“ (1%)L/m: %§2~Q2.2)1//Lf .2 $13171 s :2
334213 2 9:34 New ’ ("2/ 'WSYOL/r’r51357/
:2 Lame
F =2 4.75 R. R. 13.7 “MS '06)()I2.
4 Q56: LL, 09 0(205
L A
0.9— : A 2 M85: 47.03ng (d) (7) Compute the estimated standard error of ,31 (must give this ﬁrst). Use it to compute
a 95% conﬁdence interval for ,81. Use this interval to test H0 : ,51 S 0 vs. Ha : ,61 > 0 (i.e., that the slope is positive). AF r; u—gé‘éz“ :: :1 0.0/‘fé 1+ ' ~95 KIWIream *0 2— I
x t A 6150/0 <11" «Gr {5: 1 (*1 it ﬁalQle ' l3 = @272 i ; IMXO'OIW’ 2': 0435/ 0.277
§1hc¢ 0 (15 M” [WC/arid .454; Men's M +9224”, Hi: 62/6— 04: e025. (e) (8) Estimate the mean DC output E(y) for periods when the wind speed is 8 mph.
Construct a 95% conﬁdence interval for this mean. 95 ,g 2. 8 ‘3 .— grga4él+ 9.2271/X9 : 2.022.
’5 *— ?SZ CI «95+ 5(3) e6, X129 A 1:13 X/Se'f’t ’5 VOZQ)I2. ‘ I weave)“
2.022 1 2.176? x "WOEX 72; '* 9%«3635‘7
W '138' ([,;}g47) [1.19) (f) (6) Use above interval to say if the data support the conjecture that the mean DC output
greater than 2 is achievable when the wind speed is 8 mph? (Use the interval to
perform a test of appropriate hypotheses to answer this question. #0? ﬂ”) <2 L“ 4‘3‘3 N3.) 7 2
ghu 2 L’s Nu m M
atlas;ch ) 7J1?» 7’7) £4:
Ci/é (X z: ‘02?» W 5 (g) (6) This question concerns the 3 plots attached (see next page). Identify the plot (or plots) you would use to answer each of the questions about model assumptions stated
below. Give your answer to each question justifying your answer using the plot or
plot(s). An yes/ no answer alone will not earn any points. Questions Plot(s)
Do the errors(e’s) have a A
normal distribution? Your answer Is the variance (0?) B C
constant for all X’s? / Tdnspwmt 0% Me {ﬁn 44L ‘j—éﬂteﬂ/‘ZZM "5
Crustal/kl” ’QT JL‘W
Wm ML 76~W>cck7 514413124? {WWW Wu;
1’5 Crustal/6" nemsg ad’s, Is the straight line model b
adequate? ) ...
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 Null hypothesis, 5 m, 7 W, 35C Emwu

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