Exam 2 Practice Problems

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1 ST 312 Exam 2 Practice Problems MATERIAL COVERED ON EXAM 2 Reiland Lecture Handouts Corresponding textbook chapters 6: Infererence for the Difference Between Means Chapters 22, 23 7: Scatterplots, Correlation, Simple Linear Regression Chapters 6, 7, 8, 25 8: Transformations and Curvilinear Regression Chapter 9 9: Multiple Regression Including Indicator Variables, Interaction Chapters 28 and 29; Chapter 29 online at The above material is covered on webassign homework 5, 6, 7, and 8. NOTE: tables of the normal distribution and t-distributions will be supplied with the test. WARNING! these sample problems may not cover all topics for which you are responsible on the final exam 1. A copy machine dealer has data on the number x of copy machines at each of 89 customer locations and the number y of service calls in a month at each location. Summary calculations give x œ 8.4, s B œ 2.1, y œ 14.2, sc œ 3.8, and r œ.86. What is the slope of the least squares regression line of number of service calls on number of copiers? a b c d. none of these e. can't tell from information given 2. In the setting of the previous problem, about what percent of the variation in number of service calls is explained by the linear relation between number of service calls and number of machines? a. 86% b. 93% c. 74% d. none of these e. can't tell from information given 3. Outdoor temperature influences natural gas consumption for the purpose of heating a house. The usual measure of the need for heating is heating degree days. The number of heating degree days for a particular day is the number of degrees the average temperature for that day is below 65 F, where the average temperature for a day is the mean of the high and low temperatures for that day. An average temperature of 20 F, for example, corresponds to 45 heating degree days. A homeowner interested in switching to solar heating panels collects the following data on her natural gas use for the months October through June, where x is heating degree days per day for the month and y is gas consumption per day in hundreds of cubic feet. Month Oct Nov Dec Jan Feb Mar Apr May June x y * If ÐB BÑÐC CÑ œ *"Þ$"ß = œ "$Þ%ß = œ Þ(%, calculate the correlation coefficient r and interpret 3œ" 3 3 B C its value; draw a scatterplot of the data. 4. Each of the following statements contains a blunder. In each case explain what is wrong. a. There is a high correlation between the sex of American workers and their income." b. We found a high correlation ( r œ 1.09) between students' ratings of faculty teaching and ratings made by other faculty members." c. The correlation between planting rate and yield of corn was found to be r œ.23 bushel." 5..A study of 1,000 families gave the following results: average height of husband œbœ') 8 inches, = B œþ( in.; average height of wife œcœ'$ inches, = C œþ& in.; <œþ&. Estimate the height of a wife when her husband is ( inches tall. a. 63 inches b. 72 inches c. 64 inches d. none of these e. need more information

2 page 2 6. Outdoor temperature influences natural gas consumption for the purpose of heating a house. The usual measure of the need for heating is heating degree days. The number of heating degree days for a particular day is the number of degrees the average temperature for that day is below 65 F, where the average temperature for a day is the mean of the high and low temperatures for that day. An average temperature of 20 F, for example, corresponds to 45 heating degree days. A homeowner interested in switching to solar heating panels collects the following data on her natural gas use for the months October through June, where the explanatory variable x is heating degree days per day for the month and the response variable y is gas consumption per day in hundreds of cubic feet. Month Oct Nov Dec Jan Feb Mar Apr May June x y Summary statistics are as follows: B œ "Þ&%ß C œ &Þ&*ß = B œ "$Þ%ß = C œ Þ(%ß < œ Þ**! Calculate the least squares regression line y œ,!, " B of gas consumption y on heating degree days x. Draw the regression line on a scatterplot of the data. 7. To investigate the possible link between fluoride content of drinking water and cancer, the cancer death rates (number of deaths per 100,000 population) from in 20 selected U. S. cities - the ten largest fluoridated cities and the ten largest cities not fluoridated by were recorded. These data were used to calculate for each city the annual rate of increase in cancer death rate over this 18 year period. The data are given below: FLUORIDATED NONFLUORIDATED Annual Increase in Annual Increase in City Cancer Death Rate City Cancer Death Rate Chicago Los Angeles.8875 Philadelphia Boston Baltimore New Orleans Cleveland Seattle.4923 Washington Cincinnati Milwaukee Atlanta St. Louis Kansas City San Francisco Columbus Pittsburgh Newark Buffalo Portland a. Construct a 95% confidence interval for the difference between the mean annual increases in cancer death rates for fluoridated and nonfluoridated cities. (Use 18 degrees of freedom; do not assume equal variances). b. What assumption(s) must be satisfied for the confidence interval in part a) to be valid? 8a. We are interested in comparing the average supermarket prices of two leading colas. Our sample was taken by randomly selecting eight supermarkets and recording the price of a six-pack of each brand of cola at each supermarket. The data are shown in the following table: Supermarket Brand 1 cola $ Brand 2 cola $ Difference $ Summary statistics:. œ!þ!$(&ß =. œ!þ!$)" Find a 98% confidence interval for the difference in mean price of brand 1 and brand 2. i)!þ!$(&!þ!%!% ii)!þ!$(&!þ"$*$ iii)!þ!$(&!þ!%(" iv)!þ!$(&!þ!$%(

3 page 3 8b. Perform a hypothesis test to test if the difference in mean prices of brand 1 and brand 2 colas is different from 0. Use α =.02. Define the parameters, state hypotheses, calculate test statistic and T -value. 9. A new weight-reducing technique, consisting of a liquid protein diet, is currently undergoing tests by the Food and Drug Administration (FDA) before its introduction into the market. The weights of a random sample of five people are recorded before they are introduced to the liquid protein diet. The five individuals are then instructed to follow the liquid protein diet for 3 weeks. At the end of this period, their weights (in pounds) are again recorded. The results are listed in the table. Let. " be the true mean weight of individuals before starting the diet and let. be the true mean weight of individuals after 3 weeks on the diet Weight before diet Weight after diet Test to determine if the diet is effective at reducing weight. Use α œþ"!. 10. A simple linear regression was used to predict the score y on a final exam from the score xon the first exam. The slope of the least squares regression line is.(&. The standard error of the slope is."" and the sample size is 200. A 90% confidence interval for the true slope is (though the.0 is 8, use in the > - table) a..'% to.)' b..&$ to.*( c..&( to.*$ d..'% to.)' e. none of the above 11. Refer to the previous problem. To test the null hypothesis that the slope is zero versus the one-sided alternative that the slope is positive, we use the test statistic t œ a b..05 c..15 d..95 e If a pair of variables have a strong curvilinear (that is, nonlinear) relationship, which of the following is true? a. The correlation coefficient will be able to indicate that a nonlinear relationship is present b. A scatter plot will not be needed to indicate that a nonlinear relationship is present c. The correlation coefficient will not be able to indicate the relationship is nonlinear d. The correlation coefficient will be exactly equal to zero

4 page Sociologists are of the opinion that there has been a decrease in the difference in ages at first marriage for men and women since We want to examine data to determine if this decrease is significant. The following data summary and regression results were obtained, where the B variable is year and the C variable is the age difference (husband age wife age) at first marriage. Variable Count Mean StDev Year ( B) husband-wife age ( C) a. Interpret the value of the least squares slope, ". b. What is the value of the test statistic for testing L! À "" œ!? c. For the hypothesis test L! À" " œ! vs L+ À" "!ß select the choice below that gives the correct T- value and correct conclusion. i. The T-value is 0.68; do not reject L! À "" œ!; there is no linear relationship since 1975 between year and age difference between husband and wife at first marriage. ii. The T-value is ; reject L! À "" œ!; there is evidence that since 1975 the age difference (husband age wife age) has increased. iii The T-value is ; reject L! À "" œ!; there is evidence that since 1975 the age difference (husband age wife age) has decreased. iv The T-value is ; do not reject L! À "" œ!; there is no linear relationship since 1975 between year and age difference between husband and wife at first marriage. d. What is a 95% confidence interval for the slope? e. Find a 95% confidence interval for the mean difference (husband age wife age) at first marriage in f. Find a 95% prediction interval for the difference (husband age wife age) for a particular couple getting married for the first time in 1998.

5 page If a residual plot exhibits a curved pattern in the residuals, this means that: a. the B and C variable should be reversed so that a least squares line is appropriate for this data b. a least squares line is not appropriate because there is a nonlinear relation between B and C c. there is no significant relation between B and C d. there is a problem with the accuracy of the data Use the following Excel output to answer questions 15, 16, and 17 below. Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 8 ANOVA df SS MS F Regression Residual Total Coefficients Standard Error t Stat P-value Intercept x Which of the following is true? a. The correlation between x and y must be approximately b. The correlation between x and y must be approximately c. The correlation between x and y must be approximately d. The correlation between x and y must be approximately Which of the following is true? a. x explains about 88.5% of the variation in y b. y explains about 88.5% of the variation in x c. x explains about 78.4% of the variation in y d. y explains about 78.4% of the variation in x 17. The predicted value of C when B œ "& is approximately a b c d

6 page 6 Use the following Excel output to answer questions 18, 19, and 20 below. 18. Given this information, what percent of the variation in the C variable is explained by differences in the explanatory variable? a. About 75 percent b. Approximately 57 percent c. Can't be determined without having the actual data available d. About 25 percent 19. Given this information, what was the sample size used in the study? a. 8 b. 18 c. 9 d If the B variable increased by 2 units, then C would change by approximately a b c d e

7 page If a least squares line were determined for the data in each scatterplot, which least squares line would result in the smallest sum of the squares of the residuals? a. A b. B c. C d. D A B C D 22. For all students at Walden University, the prediction equation for Cœcollege GPA (range 0-4.0) and B" œ high school GPA (range 0-4.0) and B œ college board score (range ) is s C œ!þ!!þ&!b"!þ!!b a. Find the predicted college GPA for students having (i) high school GPA = 4.0 and college board score =800; (ii) B œ.! and B œ!!. " b. A high school student trying to gain admission to Walden University re-takes the college board exam during the summer after his senior year in high school. If he increases his college board score by 100 points, what will be the change in his predicted college GPA? Note that his high school GPA does not change since he has already graduated from high school. c. For all students with B œ &!!, what is the predicted college GPA when high school GPA is the explanatory variable?

8 page For a study of crime in the United States, data for each of the 50 states and Washington, DC was collected on the violent crime rate (per 100,000 citizens), poverty rate (percent of the population), single parent households (percent of all state households), and urbanization (percent of state population living in urban areas). The multiple regression output is shown below where Cœviolent crime rate, B " œ poverty rate, B œ single parent households, and B œ urbanization. $ R e gre ssion Sta tistics Multiple R R Square Adjusted R Square Standard Error Observations 51 ANOVA df SS MS F Signific Regression Residual Total Coe fficie nts Sta nda rd Error t Sta t P-va lue Intercept E-08 poverty single parent E-07 urbanization E-05 a. What is the least squares prediction equation for the violent crime rate? b. What is the sum of the squares of the residuals SSE for this multiple regression model? c. What proportion of the variation in the violent crime rate is explained by poverty rate, single family households, and urbanization? d. If the poverty rate increased by 1 percent, with single parent households and urbanization unchanged, how would the violent crime rate change? e. Is this model useful for predicting the violent crime rate?

9 page Over 6 decades the Gallup Organization has periodically asked the following question: If your party nominated a generally well-qualified person for president who happened to be a woman, would you vote for that person? Below is a table showing the percentage answering yes and the year of the century (37 = 1937). % Yes Year summary statistics: B œ '(Þ%% = B œ "'Þ( C œ '"Þ)" = C œ "(Þ"* < œ Þ*(" a. Determine the estimates,! and," of the parameters "! and "" in the linear model y œ "! " " x %, where B is year and C is the percentage who respond yes. b. Use the least squares line to estimate the percentage of respondents that would say yes in c. Determine the estimate = / of the standard deviation 5 of the error component % "' (note that the sum of squares of residuals ÐC sc Ñ œ ). 3œ" 3 3 = / d. Calculate a 95% confidence interval for the slope " ". (Note that WIÐ, " Ñ œ ) 8 " = B e. Conduct an appropriate hypothesis test (use α =.05) to determine if the year of the century is useful for predicting the percentage of respondents that would answer yes to the above question. State the hypotheses, find the value of the test statistic, and state your conclusion based on the P-value or the rejection region. f. What assumptions must be (approximately) satisfied for the above statistical procedures to be valid? Perform 2 diagnostics to check these assumptions. 25. Suppose you fit the quadratic model IÐCÑ œ " " B " B to a set of 8 œ! data points and found! V œ!þ*!ß ÐC CÑ œ $(Þ"%ß and WWI œ $Þ)(Þ 3œ" 3! " 25a. Is there sufficient evidence to indicate that the model contributes information for predicting C? Test using α=0.10 i) The null hypothesis is L! : a) "! œ "" œ " œ! b) "! œ "" œ " Á! c) "" œ " œ! d) "" œ " Á! e) "! œ "" œ "! ii) The alternative hypothesis is L+ À a) at least one of "! ß "" ß "! b) at least one of "! ß "" ß " Á! c) at least one of "" ß "! d) at least one of " ß " Á! e) at least one of " ß "! " " iii) What is the value of the test statistic? a) '&Þ"$ b) "'Þ% c) ($Þ!' d) %)Þ)) e) )$Þ$ iv) If the œ!þ!!&, what is the conclusion for this test? a) Do not reject the null hypothesis. Conclude that there is not sufficient evidence that the model is useful for predicting CÞ b) Reject the null hypothesis. Conclude that there is not sufficient evidence that the model is useful for predicting CÞ c) Reject the null hypothesis. Conclude that at least one of "" ß " is nonzero and that the model is useful for predicting CÞ d) Reject the null hypothesis. Conclude that both "" and " are greater than 0 and that the model is useful for predicting C. e) Do not reject the null hypothesis. Conclude that since "" œ! and " œ!, then "! must be significantly different from 0.

10 page 10 25b. What null and alternative hypotheses would you test to determine whether upward curvature exists? a) L! À" œ!ßl+ À" Á! b) L! À" œ!ßl+ À"! c) L! À" œ!ßl+ À"! d) L À "!ß L À " Á! e) L À "!ß L À "!! +! + 25c. What null and alternative hypotheses would you test to determine whether downward curvature exists? a) L! À" œ!ßl+ À" Á! b) L! À" œ!ßl+ À"! c) L! À" œ!ßl+ À"! d) L À "!ß L À " Á! e) L À "!ß L À "!! +! Consider the following prediction equation with an interaction term: s C œ $Þ' "ÞB" Þ%B!ÞB" B Þ When B is held fixed at $ß how much does the predicted value of Cchange when B" increases by 1 unit? a) 1.8 b) 10.8 c) 11.4 d) 4.2 e) Consider the printout for an interaction regression between a response variable C and two explanatory variables B and B Þ " a. Write the prediction equation for the interaction model. b. Test the overall utility of the interaction model using the global F-test at α =.05. c. Test the hypothesis (at α =.05) that B" and B interact positively. d. Estimate the change in C for each additional 1-unit increase in B when B = *. " 28. An elections officer wants to model voter turnout C in a precinct as a function of type of election, national or state. 28a Write a model for voter turnout, C, as a function of type of election. i. Cœ"! " " B %, where Bœ" if the election is national, Bœ! if the election is local. ii. C œ "! " " B" " B % ß where B" œ " if the election is national, B" œ! if the election is not national; B œ " if the election is local, B œ! if the election is not local. iii. Cœ"! " " B %, where Bœvoter turnout. iv. C œ " " B " B %, where B œ voter turnout.! " 28b. Interpret the value of " " Þ i) the rate of increase in the voter turnout for national elections ii) the mean voter turnout for national elections iii) the difference between the mean voter turnout in national and local elections iv) the difference between the mean voter turnouts in national elections 1 year apart

11 page An elections officer wants to model voter turnout C in a precinct as a function of the type of precinct: urban, suburban, or rural. The elections officer uses the following regression model: C œ "! " " B" " B % where B" œ " if urban,! if not, B œ " if suburban,! if not. Interpret the value of " Þ i) the difference between the mean voter turnout for suburban and rural precincts ii) the rate of increase in voter turnout C for suburban precincts iii) the mean voter turnout for suburban precincts iv) the difference between the mean voter turnout for suburban and urban precincts SOLUTIONS 1. sc b. since b œ r œ œ c. since r œ (.86) œ.74 ) s ÐB BÑÐC CÑ 3 3 B 3. 3œ" r œ Ð8 "Ñ= B= C œ Þ**!Þ There is a strong positive linear relationship between heating degree days and gas consumption. SCATTERPLOT: Heat. Deg. Days (x), Gas Consump.(y) GAs Consumption Heating Degree Days GAS CONSUMPT. 4. a. The correlation we are studying measures the linear relationship between 2 quantitative variables; sex is a categorical variable. b. 1 Ÿ r Ÿ 1 is violated. c. r has no units. 5. c. The husband is 4 inches, or 4/2.7 œ 1.5 standard deviations above the mean husband height. The wife's height is predicted to be above average by.25(1.5) œ.4 standard deviations, or inches œ 1 inch. (Recall b r(s /s )) œ C B = " =! " b "! " C 6., œ < ;, œ C, Bà œ.202;, œ C, B œ 1.23 B

12 page 12 Gas Consumption/Day Heating Degree Days/Day Line Fit Plot Gas Consumption/Day Predicted Gas Consumption/Day Heating Degree Days/Day a. B , s œ 2.753, n œ 10; B œ , s œ 2.429, n œ 10, t 2.101, " œ 1 " 2.025,") œ ( ) œ œ (.5962, ). b. (i) The two samples must be independently selected (ii) The distributions of increases in cancer death rates for fluoridated and nonfluoridated cities can be approximated by normal models. (iii) The two groups we are comparing must be independent of each other.. 8a. i).. > œ!þ!$(& Þ**)! œ!þ!$(&!þ!%!%þ Þ!"ß( = 8 8b. L! À.". œ.. œ!!þ!$)" ) LEÀ.". œ.. Á! α œ Þ! where. is the mean price of brand 1 cola and. is the mean price of brand 2 cola. " Þ!$(& test statistic: > œ œ Þ()%. Þ!$)" ) œ T Ð> ( Þ()%Ñ œ Þ!(Þ Since Þ!, do not reject the null hypothesis L! À.". œ.. œ!. There appears to be no significant difference in the mean price of the 2 brands of soda. 9. L! À. ". œ.. œ!ßle À. ". œ..!àα œþ"!. œ &ß =. œ "Þ&)Þ. & > œ œ œ (Þ!)à T -value œ T Ð> (Þ!)Ñ Þ!!"Þ =. "Þ&) 8 & % Conclusion: Reject L! and conclude that the mean weight of individuals after the diet is less than the mean weight of individuals before the diet. 10. c," > WIÐ, " Ñ œ Þ(& "Þ'&&Þ"", WIÐ, Ñ Þ(& Þ"" " 11. a >œ œ 12. c " 13. a.," œ!þ!% (approximately); this means that each year since 1975 the average difference (husband age wife age) has decreased by Þ!%,"!Þ!$*&'ÞÞÞ b. > œ WIÐ, Ñ œ!þ!!&&!$þþþ œ %Þ$&$"" " c. iii. The T -value given in the Excel output is always for a 2-tail test; since we are conducting a 1-tail test!þ!!!&& L+ À ""!, the T -value is œ!þ!!!"(& d. Ð!Þ!$&$'*(ß!Þ!"&%$$&Ñ from the output; notice that the interval is entirely negative. e. use sc "**) > 8 WIÐ. s"**) Ñ; for the calculations below, note that from the output we have = / œ!þ")'''%*; C œ %*Þ*!"$!%$!Þ!$*&'&Ð"**)Ñ œ Þ!$( s "**) WIÐs. Ñ œ WI Ð, Ñ ÐB BÑ = "**) / œ ÐÞ!!&&!$$"&Ñ Ð"**) "*)'Þ&Ñ!Þ")'''%* " / 8 % œ Þ!($)'))*à > 8 WIÐ. s"**) Ñ œ Þ!(%Ð Þ!($)'))*Ñ œ Þ"&$!%à so sc "**) > 8 WIÐ. s"**) Ñ œ Þ!$( Þ"&$!% Ê ("Þ))$(*', Þ"*!!%) f. use sc > WIÐC s Ñ; "**) 8 "**)

13 page 13 for the calculations below, note that from the output we have = / œ!þ")'''%*; C œ %*Þ*!"$!%$!Þ!$*&'&Ð"**)Ñ œ Þ!$(à s "**) s "**) " / = 8 / / WIÐC Ñ œ WI Ð, Ñ ÐB BÑ = œ ÐÞ!!&&!$$"&Ñ Ð"**) "*)'Þ&Ñ!Þ")'''%* %!Þ")'''%* œ Þ!!("%à > 8 WIÐC s"**) Ñ œ Þ!(%ÐÞ!!("%Ñ œ Þ%"')à so sc > WIÐC s Ñ œ Þ!$( Þ%"') Ê "Þ'!(ß Þ%&$) "**) 8 "**) 14. b 15. b 16. c 17. a 18. b 19. c 20. e 21. a 22. a. i) sc œ!þ!!þ&!%þ!!þ!!)!! œ $Þ) ; ii) sc œ!þ!þ&!þ!!þ!!!! œ "Þ'. b. predicted college GPA will increase by!.!!"!! œ!þ. c. s C œ!þ!!þ&!b"!þ!!&!! œ!þ!!þ&!b" " œ "Þ!Þ&!B" <37/ s <+>/ œ :9@/<>C <+>/ =3816/ :+</8> ?<,+83D+>398 b. SSE œ 831, c. V œ d. the violent crime rate will increase by approx per 100,000 citizens. e. analyze the model by examining the F test statistic for the F test; the hypotheses are L! À "" œ " œ " $ œ!ß LE À at least one of the " 3's is not zero. QWI '*%"()Þ% From the regression output: J œ WWI œ "(')$Þ%' œ $*Þ&&(* ; the "Significance F" (that is, the P-value) is less than Therefore, reject and conclude that at least one of the " 's is not zero. = "(Þ"* L! 3 C 24. a.," œ < = œ Þ*(" "'Þ( œ Þ***%*à,! œ C," B œ '"Þ)" Þ***%*Ð'(Þ%%Ñ œ &Þ&*$&); B b. s C *( œ &Þ&*$&) Þ***%*Ð*(Ñ œ *"Þ$' WWI &&Þ(%) c. = / œ œ 8 "% œ %Þ(%; = / %Þ(% d. WIÐ, " Ñ œ =!''"; 8 " = œþ B "&"'Þ( confidence interval is," > 8 WIÐ, " Ñ œ Þ***%* Þ"%&ÐÞ!''"Ñ œ Þ***%* Þ"%"() Ê ÐÞ)&(("ß "Þ"%"(Ñ e. L! À" " œ! vs L+ À" " Á!à," Þ***%* test statistic > œ WIÐ, Ñ œ Þ!''" œ "&Þ"à " for α œ Þ!&ß the rejection region is > Þ"%& and > Þ"%& ( 8 œ "' œ "%.0Ñ; the T -value is 0 to nine decimal places. Conclusion: since the test statistic is in the rejection region, reject L! À "" œ! and conclude that year is useful for predicting the percentage of respondents that will answer yes to the question. f. In the model C 3 œ "! " " B 3 % 3, the error terms % 3 are assumed to be independent and normally distributed with mean 0 and standard deviation 5 for all B, that is, % µ iid RÐ!ß 5Ñ for all BÞ Below is the least squares line on the scatterplot, a plot of the residuals against the -values, and a histogram B of the residuals. The first 2 plots cast some doubt on the assumption of independent errors since the residuals appear to occur in groups of positive and negative residuals. The histogram casts some doubt on the normality of the errors since the residuals show some left-skewness. 3

14 page %Yes vs Year y = x R² = Residuals Residual Plot Year 25a i) c 25a ii) d 25a iii) c Test statistic is àsince 8œ! and 5œß Jœ QWV QWI WWI $Þ)( QWI œ œ œ!þ(( Þ To determine QWVß note that since WWX œ ÐC CÑ œ $(Þ"%, and 8 5 " "( WWX œ WWV WWI (that is, Total Sum of Squares œ Sum of Squares for Regression Sum of Squares of the! 3œ" 3

15 page 15 Residuals; see Annotated Excel Regression Output in the Lecture Handouts section of our class webpage), then WWV œ WWX WWI œ $(Þ"% $Þ)( œ $$Þ(. So QWV WWV $$Þ( œ 5 œ œ "'Þ'$& and QWV "'Þ'$& J œ œ œ ($Þ!' QWI!Þ(( 25a iv) c 25b c 25c b 26 a 27a s C œ "'Þ(! $Þ!$($B" "Þ!%'&B %Þ!("(B" B "" 27b Test the null hypothesis L! À "" œ " œ " $ œ!. The J statistic is *$*% with T -value Þ"" x "! ; since the T-value Þ!&, reject L! and conclude that the model is useful for predicting C. & 27c Test L! À " $ œ! vs L+ À " $!. The test statistic is > œ *Þ"( and the 2-tailed T -value œ *Þ%) x "! ß so & the T-value for this 1-tailed test is T-value œ %Þ(% x "!. Reject L! À " $ œ! and conclude that B" and B interact positively. 27d $$Þ'!) 28a i) 28b iii) 29 i)