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stat252 wint07 soln to practice exam2

# stat252 wint07 soln to practice exam2 - STAT 252 Winter...

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Unformatted text preview: STAT 252 Winter 2007 Solutions to Practice Exam 2 1) For each of the following questions, name a method which might be used to help answer the question. Ifnone of the mahods we’ve discussed this term so far will help answer the question, you should indicate an answer 0 “not covered” Possrble choices are ANOVA, chi-square test of independence, chi-square test for a one-way table, simple linear regression. a) Do home prices differ by location? ANO VA b) Is political afﬁliation (cg Democrat, Republican, etc.) related to income level (eg. lower, middle, etc)? Chi-square test of independence c) How doesaperson’sweightdependonammmtofsugarimake? Simple linear regression d) Is there an association between gender and attendance (attend or not attend) of athletic events at Cal Poly? Chi-square test of independence e) CanSAT scorebeusedtopredictcollegeﬁeslmiangradepoimaveiage? Simple linear regression 1) Are the proportions of Cal Poly students from San Luis Obispo, Santa Barbara, and Los Angeles Counties the same? Chi-square test for a one-way table g) Do average starting salaries differby college major? ANO VA 2) The chairperson of a psychology department suspects that some her faculty are more popular with students than others. There are three sections of introductory psychology, taught at 103m, 113m, and 12pm, by professors Larry, Curly, and M0. The mimber of students who enroll in each of the sections is 32, 25, and 10, respectively. a) State the appropriate null and alternative hypotlwses necessary to determine if there is a preference for one of the professors. H0 :p1 =p2 2p3 =1/3 (mereisnopreferenceforanyprofessor) H a : At least one of the proportions exceeds 1/3. (There is a preference for one of the professors.) b) Calculate the appropriate test statistic. First ﬁnd expected counts: E(nl) = E(n2) = E(n3) = ﬂ = 22.333 3 2 _ (32 — 22333)? + (25 — 22.333)Z + (10 — 22.333)2 Test statistic: x — 211.313 22.333 22.333 22.333 c) Make a decision (using a critical value or approximate p—value), and state a conclusion in the context of the question. Assume a = .05. Critical Value: 1.35 ﬁom the chi-square distribution with dfzk—1=3—1=2.So 1,3,5 =5.99 Since 11.313 > 5.99, rejectthe null hypothesis and conclude at the .05 level that there is a preference for one of the professors. 3) Is there an association between gender and handedness? The Health and Nutrition Examination Survey took a random sample of 2237 Americans, aged 25-34 and information about gender and handedness are ’ven in the following table: Aunbhkvnnnw Illlﬂﬂliillll 20 1067 8 1170 28 Right Handed Left Handed 934 113 1070 92 2004 205 Illlllliillllll a) State the appropriate null and alternative hypotheses necessary to determine if there is a relationship between gender and handedness. H 0 : There is no relationship between gender and handedness (they are independent) H a : There is a relationship between gender and handedness (they are dependent) b) Calculate the appropriate test statistic. Note the row and column totals were added to the table. Expected oomts: E(n“ ) = W : 955.86 150,12): W 2 9m E0113): (1067)(28) : 13.36 2237 2237 2237 0 0 1 E0121) = (1_17_)([email protected] =1048.14 £0122) = WG—5) :10722 E0123) = W = 14.64 2237 2237 2237 Test Statistic: 2 _ (934—95586)2 +(113—9778)2 +(20—1336)2 1 955.86 97.78 13.36 __ 2 _ 2 _ 2 + (1070 1048.14) + (92 107.22) + (8 14.64) 21186 1048.14 107.22 14.64 c) Make a decision (using a critical value or amoxitmte pavalue), and state a conclusion in the context of the question. Assume a = .05 . Critical Value: 112,5 from the chi-square distribution with df = (r—1)(c—l) : (2—1)(3—l)= 2. So 1:15 2 5.99 Since 11.86 > 5. 99, reject the null hypothesis. Conclude at the .05 level that handedness and gender are dependent- there is a signiﬁcant relationship between gender and handedness. 4) There arefourmachines inafactorythatamsupposedtobeproducingthinmetalpipesofaﬁxed diameter. The one-way ANOVA was performed in MINITAB to determine if there was a difference intheaverage diametersofthemetalpipssforthefun machines. Tenpipesfrom each of the four machines were randomly selected and their diameters (in inches) were measured a) The following plots were constmcted for the data Brieﬂy explain how the plots support two of the mam assumptions rggniied to conduct an ANOVA F-test VIII)! g-Q—ml wwrh Muchhez ' {r >Malhne3 A-Mmhhe‘t ~: 2mm 5035 N AD 9 , uzenmnmum, omummnmow nzzornmnmucr umuoennoziuen. madman) U! From the side-by—side boxplots, we can observe that the lQR’s of all four samples are very Similar. There is no evidence to suggest that the variances of the populations from which these samples Were drawn are not equal. From the normal probability plot, we can observe that for each sample, the corresponding points fall close to a straight line. There is no evidence to sawest that these four samples were not drawn from normal populations. b) The following is an incomplete ANOVA table for the data. Fill in the rest of the table wherever you see an asterisk *: Source —_I_—_ Total c) Explain the conclusion of this ANOVA in the context of the problem. You can assume a .05 level of signiﬁcance. Since the p-value=.001 < .05, we can conclude that atieast two machines are producing metal pipes with different average diameters. d) The Tukey-Kramer output from MlNlT AB is presented on the following page, and the sample mean diameters (in inches) for each of the four imchines (l, 2, 3, and 4) are given below a = .245 :2 = .293 553 = .213 E4 = .268 Construct a simpliﬁed graph that quickly explains these results AND provide a brief interpretation. Machine 2 Machine 4 Machine 1 Machine 3 f2 = .293 @2268 32.245 £32213 From the diagram we can conclude at the simultaneous conﬁdence level of 95% that the mean diameter of the pipes produced by Machine 2 is signiﬁcantly larger than the mean diameters of the pipes produced by Machines 1 and 3. The mean (hunter of the pipes produced by Machine 4 is also signiﬁcantly larger than the mean diameters of the pipes produced by Machine 3. The mean diameter ofthe pipes produced by Machines 2 and 4 are not significantly diﬂerent, the mean diameter of the pipes produced by Machines 1 and 3 are not signiﬁcantly different, and the mean diameter of the pipes produced by Machines 4 and 1 are not signiﬁcantly diﬂ'erent. Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons Individual confidence level = 98.93% Machine 1 subtracted from: Machine Machine Machine Machine Machine Machine Machine Machine Machine Machine Machine Machine Machine (JON awa Lb 3 4 lb 3 4 n Lower Center Upper 0.00055 0.04800 0.09545 ~0.07945 —0.03200 0.01545 —0.02445 0.02300 0.07045 ———————— +—————————+—————————+—————————+— ( —————— * —————— > ( ————— * —————— ) ( ————— * —————— ) ———————— +—————————+—————————+—————————+— —0.070 0.000 0 070 0 140 subtracted from: Lower Center Upper —0.12745 —0.08000 —0.03255 —0.07245 —0.02500 0.02245 ———————— +—————————+——————-——+—————————+— ( —————— * ————— ) ( ————— * —————— ) ———————— +—————————+‘————~———+—————————+— —0.070 0.000 0.070 0.140 subtracted from: Lower Center Upper — —-— -+—- —— ———+ —————— + —— ———+~ 0.00755 0.05500 0.10245 ( —————— * —————— ) ———————— +——-——————+———————-—+——~——————+- —0 070 0.000 0 070 0.140 5) The following scatterplot is based on a random sample of 100 homes that were sold in a mid- western region of the United States Deﬁne x— — size of the home in square feet, and y— — —price of the property in thousands of dollars. Some useful MINIT AB output is given below the scatterplot: minimum ' Predictor Coef SE Coef T P Constant 9.16 10.76 0.85 0.397 size 0.077008 0.006626 11.62 0.020 Pearson correlation of price and size = 0.761 a) Based on the scatterplot, describe the relationship between size of the property and price. Moderate positive linear relationship between size of the home and its price. b) Suppose we want to predict the price of a home with the size of the home. Write out the equation of the least squares regression line. ﬁ:9.l7+.077x c) Give precise interpretations of the estimated slope and y—intercept of the regression line in the context of the problem. Explain whether or not the interpretation of the y—interoept makes sense. Estimated slope: The estimated average price of a home increases by .077(1000)=770 dollars for each additional square foot in the size of the home. Estimated y-intercept: The estimated average price of a home that is 0 square feet costs 9.17(1000)=9170 dollars. This intem'etation does notmake sense since no home is going to be 0 square feet. d) Calculate the coefﬁcient of determination and interpret its value. 2 z .7612 = .579 About5896 ofthe van'ationin tireprice ofa home canbe explainedby the size of the home. e) Conduct the appropriate test to determine if there is a signiﬁcant positive linear relationship between size of a property and price of the property. Carefully state the null and alternative hypotheses, write down the test statistic, give the p-value, make a decision, and state a conclusion. Use a = .05 . H 0 : ,6] = 0 (No linear relationship between sizeand price of a property) H a : ,61 > 0 (There isasigniﬁcant positive linear relationship between sizeand price of a property) v Test Statistic: tabs 2 11.62 (Note that the alternative is one-sided) P-value = .0202 = .01 Since p-value is less than .05, rejectthe null hypothesis. Conclude atthe .05 level that there is a signiﬁcant positive linear relationship between size of a property and price of the property. ...
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stat252 wint07 soln to practice exam2 - STAT 252 Winter...

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