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FXsoln_F08 - ADM 2304 Fall 2008 Final Exam Solutions 1(a...

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ADM 2304 Fall 2008 Final Exam Solutions 1 (a) Let p be the proportion supporting an election (ignoring the undecided). -Ho: p=.5: Ha: p>.5 -p-hat = 430/825 = .5212 -z = .0212 / sqrt(.5*.5/825) = 1.22 -z(.05)= 1.645 -do not reject null H, and conclude there were not more people supporting an election over coalition Optional alternative approach to compare two sample proportions from the same sample. If Ha: p(election) > p(coalition), then must calculate standard error accounting for non- independent sample proportions: -p1hat - p2hat = .5212 - .4788 = .0424 -std err is sqrt( .52*.48 / 825 + .52*.48 / 825 + 2 * .52*.48 / 825) and z = 1.22 We do not need to ignore the undecideds if we compare the non-independent proportions: -p1hat = 430/992 = .433, p2hat = 395/992 = .398, -std err is sqrt(p1hat * q1hat /992 + p2hat*q2hat/992 + 2*p1hat*p2hat/992) = .028933 -z = (.433468 - .398185 ) / .028933 = 1.219457 = 1.22. For some strange reason, many students calculated std err as Sqrt( p-bar * q-bar * ( 1/430 + 1/395 ). This is obviously incorrect. (b) Ho: p(W) = .30, p(Ont)=.39, p(Que)=.24, p(Mar)=.07 Observed Freq Exp.Freq = 992*p(i) Cont. To chi-square West 301 297.6 .0388 Ont 390 386.88 .0252 Que 240 238.08 .0155 Mar 61 69.44 1.0258 Total 992 992 1.1 Df. = 3 Critical value is 7.81 We do not reject the null H since 1.1 not > 7.81 Conclude the observed distribution is consistent with the population distribution.
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(c) Test and CI for Two Proportions Sample X N Sample p 1 39 135 0.288889 2 152 390 0.389744 Difference = p (1) - p (2) Estimate for difference: -0.100855 95% CI for difference: (-0.191344, -0.0103652) Test for difference = 0 (vs not = 0): Z = -2.10 P-Value = 0.036 Ho: p1=p2; Ha: p1 p2 P1-hat = 39/135 = .29, p2-hat = 152/390 = .39 Pooled proportion p-hat is (39+152)/(135+390) = .36 Std err is sqrt( .36*.64*(1/135 + 1/390)) = .048 Z = -.10 / .048 = -2.1 or 2.1 Reject null H since |z| > 1.96 for alpha = .05 Conclude that there is a difference between the proportions of BC voters and Ontario voters who supported the coalition. (d) Chi-Square Test: C1, C2, C3, C4, C5, C6 Expected counts are printed below observed counts Chi-Square contributions are printed below expected counts C1 C2 C3 C4 C5 C6 Total 1 73 75 35 168 60 19 430 58.52 45.95 26.01 169.05 104.03 26.44 3.584 18.370 3.109 0.007 18.637 2.094 2 39 19 17 152 139 29 395 53.76 42.21 23.89 155.29 95.56 24.29 4.050 12.761 1.988 0.070 19.742 0.914 3 23 12 8 70 41 13 167 22.73 17.84 10.10 65.66 40.40 10.27 0.003 1.914 0.437 0.288 0.009 0.726 Total 135 106 60 390 240 61 992 Chi-Sq = 88.701, DF = 10, P-Value = 0.000 Table above gives the missing values.
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