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# Experimental Statistics For Biological Sciences I ST 511

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GPA 3.79

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This 13 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 511 at North Carolina State University taught by Staff in Fall. Since its upload, it has received 81 views. For similar materials see /class/223982/st-511-north-carolina-state-university in Statistics at North Carolina State University.

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## Reviews for Experimental Statistics For Biological Sciences I

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Date Created: 10/15/15

Exam date and time MONDAY June 23 800 1100 Cox 206 Material covered on exam all topics covered in the latter portion of the course beginning with the geometric binomial and Poisson distributions webassign homework 6 through 11 Needed for exam 8quot gtlt llquot handwritten sheet of notes formulas calculator I will supply needed tables 1 N E A simple linear regression was used to predict the score y on a final exam from the score x on the first exam The slope of the least squares regression line is 75 The standard error of the slope is 11 and the sample size is 200 A 90 confidence interval for the true slope is though the d f is n 2 use 100 d f in the ttable a 64 to 86 b 53 to 97 c 57 to 93 d 64 to 86 e none ofthe above Refer to the previous problem To test the null hypothesis that the slope is zero versus the onesided alternative that the slope is positive we use the test statistic I c 15 d 95 e The failure rate or hazard rate hI of a system is related to the conditional probability that the system will fail in the next instant of time given that it has survived to time I The hazard rate of a particular brand of personal computer is thought to be a power funcition of its age I so that hI cI where c and d are parameters with unknown values that need to be estimated To convert the above expression to a linear equation involving 0 and d we take the log base 10 of each side to obtain log hI log 0 dlog I or y o 51x where y log hI 60 log 0 61 d and x log I A large number of computers were put on test for 10 hours to obtain an estimate of the hazard rate for each of the ten hours The results are shown below 110135 xi 10g1oli yi 10g10hi 1 00000 002119 2 03010 059600 3 04771 091381 4 06021 124551 5 06990 139794 6 07782 157978 7 08451 169020 8 09031 177085 9 09542 192942 10 10000 198900 The following straight line model was proposed y 50 7136 6 Summary statistics for the above data are n 10 3 35598 s 31836 g 131337 5 63369 7quot 9992 Ex 65598 2y 131337 He 9quot 0 7 2x 52152 D 208634 104295 SSXX 9121 SSyy 36149 SSXy 18141 a SET UP the calculation of 0 and b1 the least squares estimates of the model parameters 60 and i b SET UP the calculation of an estimate of 02 the variance of the random error 6 component of the mo e c Calculate the coefficient of determination and interpret its value For the remainder of the problem you may assume that the following values have been calculated 0 7 0087 b1 1989 s 0275 d SET UP the calculation of a 95 confidence interval for 61 Suppose the confidence interval is 19226 20554 Does it appear thatx log I is useful for predictingy log 111 e SET UP the calculation of a 95 prediction interval for y whenx 5 f What assumptions are required for the valid use of the confidence interval procedures in parts d and e Because there are no regional or national standards it is difficult for university admissions committees to compare graduates of different high schools An 80 average at a high school with low academic standards may be equivalent to a 70 average at a high school with high academic standards In an effort to campare applications a pilot study was conducted Random samples of students who were admitted the previous year from four local high schools were selected All of the students entered the university39s business school with high school averages between 70 and 80 Their university freshman year GPA39s were recorded The results are summarized in the table below a Is there evidence of a difference in the freshman year mean GPA39s for the four high schools Fill in the ANOVA table to determine the answer use 105 and Excel to find F critical AN OVA table Source of Variation SS df MS F P value F critical Between Groups 28336 0437 Within Group b Find an appropriate confidence interval for the difference in mean freshman year GPA for high school A and high school C 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 Y 84 Sr 21 7 142 sy 38 and r 86 What is the slope of the least squares regression line of number of service calls on number of copiers 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 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 109 between students39 ratings of faculty teaching and ratings made by other faculty membersquot c The correlation between planting rate and yield of corn was found to be r 23 bushel 8 A study of 1000 families gave the following results average height of husband Y 68 inches s1 27 in average height of wife 7 63 inches sy 25 in r 25 Estimate the height of a wife when her husband is 15 standard deviations above the mean husband height a 63 inches b 72 inches c 64 inches d none of these e need more information 9 Shown below is a scatterplot with the corresponding least squares line Scatte rplot 4 n 6 2 Y 1 v v e 1 2 7 9 3 5 O 6 01 n X Choose the residual plot that corresponds to the above scatterplot and least squares line a l c Ill IV e none I H a 5 o4 n 2 2 In A Q g 0 g 4 7 o 9 72 7 o A 1 E 1 3 71 1 3 5 7 39 X x III W m 2 7 o 9 4 7 a n v o 3 1 2 O 39a 2 7 3 2 7 0 E 1 9 n 04 7 n a 0 71 1 3 5 7 X 71 1 O 1 1 H 19 L H gt A certain hotel chain finds that 20 of the reservations made will not be used a What is the probability that among fifteen reservations made that more than 8 but less than 12 will be used b If four reservations are made what is the probability that less than two will be unused The taste test for PTC phenylthiourea is a favorite exercise for every human genetics class It has been established that a single gene determines the characteristic and that 70 of the American population are quottastersquot while 30 are quotnontasters 1f 20 people are randomly chosen and administered the test a give the probability distribution of x the number of quotnontasters out of the 20 chosen b Find P3 ltx lt 9 c Find the mean of x d Find the variance of x An oil firm is to drill twenty wells with each well having probability 02 of successfully producing oil It costs the firm 20000 to drill each well A successful well will bring in oil worth 750000 a Find the firm39s expected gain G from the twenty wells b Find the variance of the firm39s gain Ford claims that 40 of their customers are return buyers ie they have purchased a Ford product previously Suppose we sample 10 recent purchasers of a Ford product and define X of return buyers a Find P1 S x lt5 b Find Px gt 6 c What would you conclude about Ford39s claim if you found 9 return buyers in the sample and why Identify the statement that describes a situation that can be modeled as a binomial experiment a a shopping mall is interested in the income levels of its customers and is taking a survey to gather information b a business firm introducing a new product wants to know how many purchases its clients will make each year c a sociologist is researching an area in an effort to determine the proportion of households with male quotheads of household d a study is concerned with the average hours worked by teenagers who are attending high school Which of the following is a correct statement concerning the Central Limit Theorem CLT a The CLT states that the sample mean x is always equal to p b The CLT states that for large samples the sampling distribution of the sample mean is approximately normal c The CLT states that for large samples sample mean x is equal to p d The CLT states that for large samples the sampling distribution of the population mean is approximately normal e Both c and d are correct The amount ofmoney spent on food per week by a typical American family is known to have a mean of 92 dollars with a standard deviation 0 of 9 dollars Suppose a random sample of 81 families is taken and the sample mean is calculated a Describe the sampling distribution model of the sample mean Include the mean standard deviation and type of distribution if known b Find the probability that the sample mean does not exceed 904 dollars In a learning experiment untrained mice are placed in a maze and the time required for each mouse to exit the maze is recorded For untrained mice the average time to exit the maze is u 50 sec and the standard deviation is a 16 sec 1f 64 randomly selected untrained mice are placed in the maze and 00 L O N O N 1 the time necessary to exit the maze recorded for each one what is the probability that the sample mean differs from 50 by more than 3 During the lunchhour rush at a McDougalds restaurant orders for the Big Mac hamburger follow a Poisson distribution and occur at a rate of 4 per minute What is the probability that 5 or more Big Macs will be ordered in one minute Let X be the number of accidents per week at a hazardous intersection X varies with mean 22 and standard deviation 14 X takes takes only nonnegative wholenumber values so the distribution of X is certainly not normal a Let X17 X2 and X3 be the number of accidents in each of 3 different weeks at this intersection The X gs are independent and identically distributed each with the same distribution as X What is 00919919 the standard deviation of the sum X1 4 X2 4 X3 b Let Ebe the mean number of accidents per week at the intersection during a year 52 weeks What is the approximate probability that is less than 2 c What is the approximate probability that there are fewer than 100 accidents at the intersection in a year True or false In a hypothesis test a Pvalue of 003 means that there is only probability 003 that the null hypothesis is true In a Risk Management Study on fires in compartmented fireresistant buildings the data below was generated The data in the table give the number of victims who died trying to evacuate for a sample of 14 recent fires NUMBER OF FIRE VICTIMS Las Vegas Hilton Las Vegas 5 Inn on the Park Toronto 5 Westchase Hilton Houston 8 Holiday Inn Cambridge Ohio 10 Conrad Hilton Chicago 4 Providence College Providence 8 Baptist Towers Atlanta 7 Howard Johnson New Orleans 5 Cornell University Ithaca 9 Westport Central Apartments Kansias City MO 4 Orrington Hotel Evanston IL 0 Hartford Hospital Hartford CT 16 Milford Plaza New York 0 MGM Grand Las Vegas 36 Source Macdonald J N Is Evacuation a Fatal Flaw in FireFighting Philosophy RiskManagemem Vol 33 No 2 Feb 1986 p 37 14 14 gm 117 2351 if 1039214 3971 i1 1 We would like to calculate a confidenced interval for the mean number p of victims per fire a State the assumption in terms of the problem that is required for the confidence interval technique to be valid b Construct a 98 confidence interval for the true mean number of victims per fire who die attempting to evacuate compartmented fireresistant buildings c Interpret the interval constructed in part b N N N N N N N Which of the following statements is false a The t distribution is symmetric about zero b The t distribution is more spread out than the standard normal distribution c As the degrees of freedom get smaller the tdistribution39s dispersion gets smaller d The t distribution is moundshaped The Student t distribution approaches the normal distribution as the a degrees of freedom increase b degrees of freedom decrease c sample size decreases d population size increases The statistic 3 E has the student t distribution only if the sample is drawn from a Student t distribution b a normal distribution c a negatively skewed distribution skewed to the left d a positively skewed distribution skewed to the right Domino39s Pizza in Big Rapids Michigan advertises that they deliver your pizza within 15 minutes of placing an order or it is free A sample of 25 customers is selected at random The average delivery time in the sample was 13 minutes with a sample standard deviation of 4 minutes a Perform a hypothesis test to determine if we can conclude that the population mean is less than 15 minutes b What is the required condition of the technique used in part a c Approximate the Pvalue for this test A marketing consultant was interested in estimating the mean daily consumption of soft drinks among teenagers A random sample of 61 teenagers were asked how many ounces of soft drink they consume daily The sum of the observations and the sum of the squared observations are shown below 61 61 in 1365 and 2xi E2 1605328 i1 i1 Estimate with a 99 confidence interval the mean daily consumption of soft drinks by teenagers As the sample size n increases when the confidence level is held fixed the width of the confidence interval for the population mean tends to a increase b decrease c stay the same A random sample of 49 data values had sample standard deviation 5 82 Suppose the confidence interval for u calculated from the data was 63 lt u lt 69 What is the sample mean x 29 A confidence level of 99 can correctly be interpreted to mean that a 99 of the time in repeated sampling intervals calculated using an appropriate formula will contain the sample value b 99 of the time in repeated sampling intervals calculated using an appropriate formula will contain the relevant population parameter c 99 of the time in repeated sampling intervals calculated using an appropriate formula will contain the sample value as the midpoint of the interval d 99 of the time in repeated sampling intervals calculated using an appropriate formula will contain the sample mean as their midpoints 30 Suppose that 9 observations are drawn from a population whose distribution can be described by a L L N L L L Lquot normal model The observations are 15 9 13 11 8 12 11 7 10 Note that E 1067 and s 25 You want to test whether the mean of the population from which this sample was taken is different from 12 a State the null and alternative hypotheses b Compute the value of the test statistic c Approximate the P value d What is your conclusion Recently there have been campaigns encouraging people to save energy by carpooling to work Some cities have created carpool only traffic lanes i e only cars with 2 or more passengers can use these lanes In order to evaluate the effectiveness of carpool only lanes toll booth personnel in one city monitor 2000 randomly selected cars in 1999 before the carpool lanes were established and 1500 cars in 2003 after the lanes were established The results are shown below where x1 x2 is the number of cars with 2 or more passengers in the data for 1999 2003 Use a 95 confidence interval to determine whether the data indicate that the fraction of cars with carpool riders has increased over this perro 1999 nl 1500 x1 57639 2003 ng 2000 x2 652 The lower limit of a confidence interval at the 95 level of confidence for the population proportion if a sample of size 200 had 40 successes is a 2554 b 1446 c 2465 d 1535 Perform the hypothesis test shown below H0 p 05 Ha p lt 05 given that a random sample of size 1000 revealed that the number of successes was 40 Compute the Pvalue and use it to make a conclusion concerning the hypothesis test A professor claims that 70 of College of Business graduates earn more than 45000 per year In a random sample of 300 graduates 195 earn more than 45000 Perform a hypothesis test to evaluate the professor39s claim To investigate the possible link between uoride content of drinking water and cancer the cancer death rates number of deaths per 100000 population from 19821999 in 20 selected U S cities the ten largest fluoridated cities and the ten largest cities not fluoridated by 1999 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 Am xmeeee m m 1mm m cm CzrcexDeaLhRak Cw CzrcexDeaLhRak creme 106w 1m melee ms Mhhu Nu Emma 17m 54mm 21115 New 011m mm Clawbrd 194m 3 m Washm 32772 cmm ms Mawmks em Admu 4 mo Mm om Kansasc y 22132 mrmm 12275 Cumulus 17m Pmsbmgh mm eaAdi 75576 m was mm mm Cansuuct a 95 can dence meme m me difference bemenme mm annual marusesm me death mes fax uanda zd and nm uandaled cmes mm mamas mLhe me aeeumh m uundated cmes mullet p be n me cancer death me m nan uandated emes Pexfarm 5 Lamb me mm m mm amualmcreasex wahm me mph absemuansn 9 w a 7 6a a 393 SE a a a 5 e en a 3 2 a W 0 7 a 2a 0 a e Samph m Samph a2 Samph m Samph b2 enzyme dizgnmh mph u rm papmuanx s 710 A mphm Flam papu ahanl 2 a 2 21 apulauanl S13 7 s samplebl Flam papu ahanl 5 5 5 A 5 Assume um me We mphs xepxesentmdependenL nndam mphs atreatmentsmacampletelyrmdamxzed deag mph 52 mm p Refer m Pmblem 35 emeepanamg w W rmphsx mazmbam ammgms amps 556 m ehh dal diagam ehh mph measuemem and me a calm 39hztxemnentmzansx e Lhemunsa b Useme mmm calculate 39theanSquare m e cueuhhme m afsquared a exeneesbemeen Sum uquuares fax Exmx SSE fax ehh am aegm 7 thath whatpercenmge arm Tutsi 5m aquuaresstheSu39n aquuaresfm Gmups 7m ehh c5527 apprupnatenurnher ufdegees uffreednm Ca1eu1atetheF 7 Elm urthe Man Square fur GmupsMSG m the Man Square furEn ur VKSE rur ach uut magarn eqmlmmns Usea ns assure the vahurty urthe F4595 eunuueteu m harm y mahlexs the age urrraeneemuuhanu age 7 WAfeage at rst rnarnage Variable chum an Sszv Yam 4 19865 7171 husbandrme age 1 24 2 3125 n 249 a Interpret the Wine ufthelast squares sl Wha the value urthe test stausue fur Fur the hypu39hesxs test H 5 0 vs eurreetywue and eurreet cundusmn he Pmlue 15 n 58 an nut rq eet H 5 ere nu hnmr relatrunshrp snee 197s belweenytzar and age dt wzmtz belwem husband and me at rst namage u The Framers n unnlsz rejeetHn 5 a there rs emueneethat anee 197s theage dl mce stand age 7 WAfe age has mtrased m Theywuers n nunmsmeetyn 5 0Lherexs smdmcethalsmce lWSLheage dl mce hustand age 7 WAfe age has demsed upe m tesung H 5 z m H 51lt a seleet the chums heluwthat gves the xv belweenytzar and age dt wzmtz belwem husband and me at rst namage Whatrsa 95 cun dmce tram fur the stupev m1998 gemngrnarneu furthe rslumem 1998 39 Over 6 decades the Gallup Organization has periodically asked the following question If your party nominated a generally wellquali edpersanfarpresident 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 Yes92827880767366535755 55248 Year 998784837875716967635958 54 4 37 summary statistics 5 6744 SI 167 g 6181 sy 1719 7quot 971 a Determine the estimates 0 and 1 of the parameters 60 and 61 in the linear model y 60 1x e where x is year and y is the percentage who respond yes b Use the least squares line to estimate the percentage of respondents that would say yes in 1997 c Determine the estimate se of the standard deviation 0 of the error component 6 16 note that the sum of squares of residuals i2 255748 139 d Calculate a 95 confidence interval for the slope 61 Note that SEb1 7amp5 e Conduct an appropriate hypothesis test use a 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 1 c 61 itSEb1 2 a t 3 a b1 9992 b0 131337 12166698 b s2 SSEn 2 ssyy 121551ny 36149 b1181418 c 72 99922 998439 if r2 9984 then 9984 of the variation in loghl about its mean is explained by the linear relationship with log I d 1989 i 23060275 9121 Since the confidence interval for the slope is entirely positive it appears that the slope 61 is not 0 and thatx log t is useful for predictingy log 171 e itSE5 5 0087 198945 10032 1 230608df 2 2 SE29 7 Oi 3331335 92892 90083 SE5 SE2b1 gtlt x W 55 sg 00083 gtlt 6 666982 02 00076 0293 itSE5 10032 i230600293 10032 i 0676 f e N NO a for all x and are independent across observations ANOVA table Source of Variation SS df MS F Pvalue F critical Between Groups 4309506 3 1436502 28336 0437 27233 Within Group 39035948 77 506957 Reject H0 11A p5 pg 11D and conclude that at least 2 of the mean freshman year GPA39s er b Bonferroni confidence inerval for 11A pg g 0083 tooggm 2449 6 Sp 1506957 712 688250 620125 i 2449712 2 10 a 68125 i 58485 0964 12661 61 rC i 86 155 5 r2 852 74 7 a The correlation we are studying measures the linear relationship between 2 Quantitative variables sex is a categorical variable 1 S r S 1 is violated c V has no units 8 c The husband39s height 15 as standard deviations above the mean husband height of 68 inches The wife39s height is predicted to be above average by 2515 4 y standard deviations or 4 gtlt 25 inches 1 inch Recallb rsysz 9 c 111 10 a p 08 P8 ltx lt 12 0334 b p 02 Px lt 2 08192 11 a px xl gfx 03X072 X x 01220 b 078 c 5 d 42 12 Let X of successful wells XNbinomial20 02 Gain 750000X 2020000 a EG 750000EX 400000 7500004 400000 2500000 b VarG 7500002VarX 7500002200208 a 0627 b 055 c Have reason to doubt Ford39s claim since the probability that x 9 is approximately 002 when p 4 14 c 15 b 16 a the sampling distribution model is N92 798 1 b Pz S 16 00548 PE lt 47 or 37gt 53 01335 PX 2 5 1 PX 4 1 6288 3712 21 Since the X95 are independent VarX1 4 X2 4 X3 VarX1 VarX2 VarX3 588 SO 7811977219773 x 588 2425 b By the CLT E N approx N227 194 so PElt2 13ng lt 23342 Pz lt 103 1515 c Ptotal accidents lt 100 PE lt PE lt 1923 Pz lt 143 0764 false a The distribution of the population of the number of victims who attempt to evacuate fires can be approximated by a normal mo e b From the 14 observations 5 836 s 894 confidence coefficient 98 a from the I table 2 25503 13 df y i 2 836 i 2550387341 836 i 533 2031459 We are 98 confident that the interval 203 1469 contains the true mean number of victims who die attempting to evacuate compartmented fireresistant buildings 22 c 23 a 24 b 25 21 H0 u 157 Ha u lt 15 Test statistic t 250 25 b The distribution ofpizza delivery times can be approximated by a normal model 25 6 Approximate P value the test statistic t 25 is between 24922 and 27970 The P value is between 005 and 01 T183 P value 0098 Conclusion Reject the null hypothesis Yes we can conclude that the mean delivery time is less than 15 minutes 25 s 4160650328 517 a 1th 75 2238 i 2660357 2238i 176 2052 2414 27 b decrease 28 66 29 b 305 H0112 51415412 bE1067 s 25 t M 1596 UI H W HHH W90 NN rIc 0 9 c Use if distribution with 8 degrees of freedom from the ttable Pt lt 1596 Pt gt 1596 is between 01 and 02 d Do not reject H0 there is no evidence that u differs significantly from 12 31 7A m 384p 652 32 1500 2000 39 32 34 W UI W a VI 39l 384 326 i 19649 954321 Fi l 058 i 032 s 026 09 Conclusion Conclude that the proportion of all cars with carpool riders has increased over the 1999 2003 period b 33 test statistic z 1451 Pvalue Pz lt 1451 0734 Since the Pvalue is greater than 0705 do not reject H0 there is no evidence that p differs significantly from 5 195 300 H0 1 77 Ha 1p 7K 7 65 test statistic z 7 m a value Pz lt 189 Pz gt 189 0588 Since Pvalue gt 05 do not reject the null hypothesis there is no evidence that p differs significantly from 7 189P El 22573sf 2753n1 10 962 13411s 2429n2 10 a use degrees of freedom minn1 17 n2 1 min9 9 9 tist 22622 22573 13411 i 2262 9162 i 1628 7118 25442 Note that this interval will differ from the interval given on the Tl83 and Tl84 since the calculators use different degrees of freedom bH0ilL1 20Hai 1 test statistic t 2 2573531370 127 Pvalue Pt gt 127 use degrees of freedom 10 10 minn1 17 n2 1 min9 9 9 to approximate the P value from the ttable Pvalue Pt gt 127 is between 01 and 015 Therefore do not reject the null hypothesis there is no evidence that the mean annual increase in cancer death rate is significantly greater for flouridated cities than for nonflouridated cities In diagram 21 the difference between the sample means is small relative to the variability within the sample observations a a Y1 2 Y2 5 b Y1 2 Y2 5 b overall mean in diagram a overall mean in diagram b 5 35 diagram a SSG 5151 32 5252 5V 52 352 55 352 225 diagram b SSG 11151 E2 11252 E2 52 352 55 352 225 c a16 22 1 220 224 221 22 34 a2 8 52 1 523 527 526 52 34SSE343468 b12 223 222 222 221 222 b2 5 525 525 524 526 52 2SSE224 d diagram a SSTotal 225 68 905 22 5 2486 m diagram b SSTotal 225 4 265 8491 e diagram a MSG 225 MSE fag 7E2 6 88 85 F M sg 265 diagram b MSG 225MSE 7 g 5F 45 f diagram a do not reject H0 diagram b reject H0 F1 8532 g i the probability distribution in each population is normal ii the population variances are equa iii the samples from each population are random and independent 21 b1 0024 approximately this means that each year since 1975 the average difference husband age wife age has decreased by 024 b r 53m w 435311 c iii The Pvalue given in the Excel output is always for a 2tail test since we are conducting a 1 tail test Ha 61 lt 0 the Pvalue is 0 000255 00001275 d 003536977 001254335 from the output notice that the interval is entirely negative 9 use 1998 i t272SE 1998 for the calculations below note that from the output we have se 018662649 1998 4990213043 00239565221998 2037 SE 1998 SE2b1 gtlt a 32 0055033152 gtlt 1998 198652 W 07386889 1ZSE51998 2074 07386889 153204 so 1998 i 1ZSE51998 2037 i 153204 1883796 2190204 7 i a b17751 f use 1998 i t72SE1998 for the calculations below note that from the output we have se 018662649 1998 4990213043 00239565221998 2037 A 2 SElt91998gt SE2Z 1gtlt Iv 902 53 0055033152 gtlt 1998 198652 W345 0186626492 200714 1ZSE1998 2074200714 41628 so 1998 i 1ZSE1998 2037 i 41628 162072 245328 971 99949 b0 3 b1 6181 999496744 559358 b 397 559358 9994997 9136 SSE 7 255748 7 Mn 7 4774274 5 4274 7 d SEb1 Him 7 0661 confidence interval is 1 i tZSEb1 99949 i 21450661 99949 i 14178 a 85771 114127 e H0610 vs Ha 17 0 test statistict 533171 90969319 1512 for a 057the rejection region is t gt 2145 andt lt 2145 n 2 16 2 14df the Pvalue is 0 to nine decimal places Conclusion since the test statistic is in the rejection region reject H0 61 0 and conclude that year is useful for predicting the percentage of respondents that will answer yes to the question c 59

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