QUESTION 4 - problem-type question requiring detailed written answers.The members of a sports club were concerned about the effect of the weather onthe outcome of their cricket matches. The results for the 75 matches played wererecorded in the following table:ResultsWinDrawLoseTOTALGood179733WeatherBad9102352If a match was selected at random, what is the probability that:(a) The sports club did not draw the match.(b) The weather was good and the club at draw the match.(c) The club lost the match given that the weather was bad.(d) The weather was good or the sports club win the match.TOTAL26193075(e) Are the events "Win" and "Lose" the match, mutually exclusive events. Usethe rules of probabilities to justify your answer. z0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.51.61.71.8202.12.22.32.42.52.62.72.82.93.050005398579361793.23.36554691572577580788181598413864388499032919293329452955496419713977298219861989399189938995399659981998700-3.4 00030005-3.20007-3.1 0010-3.0-2.9 0019-2.8 0026-2.7 00350047-2.5 0062015040543858326217-14 0808-1.3 0968-1.2 1151-1.1 1357-1.0 158765916950-0.9 1841-0.8 2119-0.7 2420-0.6 2743-0.572917611791081868438-0.4 3446-0.3 3821-0.2 4207-0.10.01 5000866588699049920793459990999399959997This is Table IV of Appendix C.94639564964997199778982698649896992099409955996699759982998799919993999599970100030005000700090013-1.9 0287 0281-1.8 0359-1.7 044600180025003400450060-1.6 0548 0537-1.5 06680655035104360793095111311335156218142090238927093050340937831024168508049605478587162556628698573247642793982128461868688880082 008000780132-2.3 0107 0104 0102-2.2 0139-21-2.0 022801360174017901700222021790669222935794749573965697269783983098689898Table IV Standard Normal Distribution Table992299419956996799769982998799919994999599971002000100050006000900180024003300440059027403440427052606430778093411121314153917882061235826763372374545224920003512055175910629366647019735776737967823884858708890790829236937094849582P(1) ---xC₂Poisson probability formala P(x)9834987199019925994399579968997794959591966496719732 973897889793998399889991999499969997normal curve to the left of z with thevalues of z equal to 0 or positive.0300030004000600090012001700230032004300570075009901290166021202680336041805160630076409181093129215151762203323273336370744834880Joint probability of two mutually exclusive events:PA and B) 0. Addition rule for mutually nonexclusive events:P(AB)=P(A) + P(B)- PA and B)Addition rule for mutually exclusive events:P(AB)=P(A) + P(B)Chapter 5. Discrete Random Variables and TheirProbability DistributionsMean of a discrete random variable x:=xP(x). Standard deviation of a discrete random variable x:a-√2+¹P(1)-²045160555759486331Chapter 6. Continuous Random Variablesand the Normal Distribution2 value for an x valueValue of x when . o, and are known: x=+507054Chapter 7. Sampling Distributions- Mean of :. Standard deviation of I when n/Ns .05 aya/VI-K738977047995826485088729892590999251factorial: w-1-2) 3-2-1Number of combinations of a items selected x at a time:F.x-x). Number of permutations of items selected x at a time:0400030004000600080012(-x)Binomial probability formulac P(x)-C, pqMean and standard deviation of the binomial distribution:pandayHypergeometric probability formula:9959996999779984001600230031004100559992999499969997007300960125xt. Mean, variance, and standard deviation of the Poisson probability distribution:=A²A, and VĀ0207026203290409050506180749090110751271149220052296261133003669The entries in this table give thecumulative area under the standardnormal curve to the left of z with thevalues of z equal to 0 or negative.4443983898429875 9878990499069927994548400055199559659876368673670887422773480238289853195059599967897449798992999469960997087498770894489629115 9131926592799394940699849989999299949996999705000300040006000800110016002200300040005400710094012201580202025603220401049506060735088510561251146917111977226625782912326436320652394404480156.36602664066772712374547764805195159608831583408554 8577968697509803996199719985999299949996999706984698509881988499099911993199329945 9949000300040006000001100150021003900520091011901970250031403920485059407210869103814461685194922362546322835944364186752793675606464434761715774677948078of89809147929294189525961696939756980899629972997999859995999699971070003000400050008001100150021002800380051008901160150019202440307038404750582070808531020142331923557108085319571461036480719075177823810683658599- Margin of error of the estimate for aE20 of th- Determining sample size for estimating881089979162930694299535962598549913993499519963997399560003000400050007969997069761 9767981299939995999600140020where=1/√ñ0037004900660087011301460188023903010375pt 2 whereMargin of error of the estimate for pE-2 where - VDetermining sample size for estimating p-²/E²04650571069408381003140146815359575361416517687975498133838988309015Confidence interval for p for a large sample:V/n917793199545Chapter 9- Hypothesis Tests about the Meanand Proportion9857989099169936995299649986999099939995999799980900020003000500070010166016351922189422062177 214825142483 24512843 2810 2776001900480064008431563520389738594325 4286424747214641(continued on next page)0110014301830233029403670455055906810823098511701379- Population proportion: p-X/NSample proportion: -x/- Mean ofp-pStandard deviation of when a/N05: Vpq/n16111867Chapter 8- Estimation of the Mean and Proportion- Point estimate of i.Confidence interval for using the normal distributionwhen is known31213483try where aya/VConfidence interval for a using the r distribution when or isPage 11 of 13Test statistic z for a test of hypothesis about ja using thenormal distribution when is know2where yTest statistic for a test of hypothesis about ja using the t distribution when or is not known-Test statistic for a test of hypothesis about p for a largesample:-- where -Page 9 of 13M8 38333 88889 30212 9548A HAAaa aaaaa3.13.3003000539837936179655469157257758086438849903291929332945295549641971397727881791081.59 81868413 84389861989399189938995399659974019981998730403438583262176591695072917611366588699207934594639564964997199876986498969920994099559966997599829987999099939995999599979997This is Table IV of Appendix C.--ww99930025080Within-samples sum of squares547858716255698573247642793982128461Total sum of squares:972697839830Chapter 12- Analysis of VarianceLet989899229941the mumber of different samples(orta)99569967997699829987. Cumulative percentage99919994999584858686 87088888814179066 90829222 92369357 937094749573032²-.Confidence interval for the population variance oX-toTest statistic for a test of hypothesis aboutNTest statistic for a goodness-of-fit test and a test of inde-(0-15120Zury- (Emyp551762932004the size of sample7,- the sum of the values in sample ithe number of values in all samples735776737967- ++the sum of the values in all samplesthe sm of the squares of values in all samplesFor the F distributionandnormal curve to the rest of z with thevalues of z equal to 0 or positive.94849582966497329788Degrees of freedom for the somerator--1Degrees of freedom for the denominator-kBetween-samples sum of squares- Range - Largest value - Smallest value. Variance for ungrouped data(Ex)²IP-9834987199019925SST SSB SSW-1Variance between samples: MSB SSB/A-1)Variance within samples: MSW-SSW/(-A)Test statistic for a one-way ANOVA lestFMSB/MSW995799689977998399k9991999499969997Chapter 13 Simple Linear RegressionSimple linear regression model: y= A + B + eEstimated simple linear regression model: 9-a + bxPrem S. MannChapter 2- Organizing and Graphing Data. Relative frequency of a class f/2fN and ²-a² m- Standard deviation for grouped data:(Emy)Smy-N. Percentage of a class (Relative frequency) x 100. Class midpoint or mark (Upper limit +Lower limit)/2. Class width Upper boundary - Lower boundary. Cumulative relative frequency04- (Cumulative relative frequency) x 100Chapter 3 - Numerical Descriptive Measures- Mean for ungrouped data:Ex/N and 7-x/mMean for grouped data:Emf/N and I Σmf/nwhere is the midpoint and / is the frequency of a classMedian for ungrouped dataValue of the middle term in a ranked data set2² (+)²5100555759486331670070547389770479958264850887298925Cumulative frequencyTotal observations in the data set92519382-94959591967197389793andwhere is the population variance and is the sampleStandard deviation for ungrouped data(Ex)98389875990499279945In'y- (Imp)?99599900999299949996999799849988Nand s#-1where and s are the population and sample standard deviations, respectivelyVariance for grouped dataEmp05519955963987636867367088-VChebyshev's theorem:For any number & greater than 1, at least (1-1/2) of thevalues for any distribution lie within k standard deviationsof the mean742277348023828987498944911592659394950596789744979898429878990699299946063239363660266772712374547764805183158554877089629131927994069515960896869750980398469881990999319948996199609970997899849989 998999929994999699979992999499969997Sum of squares of xy, x, and55,-2-(X)17327936756064644368087157.Confidence interval for92, wherePrediction interval for74867794807883408577KEY FORMULASIntroductory Statistics, Sixth Edition8790398099629972997999799985 99859989929295259616969397569884991199329949999299959996999733195714610364300868417190751778238106836585998810www9162930694299533960976198129887991399349951Least squares estimates of A and B-55/55, and any-AStandard deviation of the sample en99619973998099869990www.5999599969997Error sum of squares SSE-1-2(y-99(₂)Total sum of square SST-1²-Regression sun of squees SSRSST-SSE. Coefficient of determination: -55/55.Confidence interval for Bbt, where 44/V55,Test statistic for a test of hypothesis about BLinear correlation coefficientVSS, 55Test statistic for a test of hypothesis about po(₂-19995359575361416517Interquartile range: IQR-0-0- The Ath percentile:722475497852813386219015917793199545970697679817Chapter 4- ProbabilityClassical probability rule for a simple event:.9916993699529964Chapter 14- Multiple RegressionFormulas for Chapter 14 along with the chapter are on theWeb site for the text.99909993Chapter 15. Nonparametric MethodsFormulas for Chapter 15 along with the chapter are on theWeb site for the text.9998Empirical rule:For a specific bell-shaped distribution, about 68% of the observations fall in the interval (-a) to (+), about95% fall in the interval (20) to (+20), and about99.7% fall in the interval (x-30) to (+30)-Q,- First quartile given by the value of the middle termamong the (ranked) observations that are less than themedianQ-Second quartile given by the value of the middle termin a ranked dataPage 10 of 13Q-Third quartile given by the value of the middle termamong the (ranked) observations that are greater thanthe medianP₁-Value of the 100th term in a ranked data set- Percentile rank of aNumber of values less than 100Total number of values in the data setPE) Total number of outcomesClassical probability rule for a compound eventPA) Number of outcomes in ATotal number of outcomes- Relative frequency as an approximation of probability:P(A) =Condition for independence of events:P(A)=P(AB) and/or P(B)=P(BA)For complementary events: P(A) + PA) 1Multiplication rule for dependent events:P(A and B)- P(A) P(BA)Multiplication rule for independent events:P(A and B)=P(A) P(B). Conditional probability of an eventP(A and B)P(A and B)P(A/B)= P(R) and P(4) P(A)Page 8 of 13

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UESTION 4 - problem-type question requiring detailed written answers. he members of a sports club were concerned about the effect of the weather on e outcome of their cricket matches. The results for the 75 matches played were corded in the following table: Results Win Draw Lose TOTAL Good 17 9 7 33 Weather Bad 9 10 23 52 TOTAL 26 19 30 75 If a match was selected at random, what is the probability that: (a) The sports club did not draw the match. (b) The weather was good and the club at draw the match. (c) The club lost the match given that the weather was bad. (d) The weather was good or the sports club win the match. (e) Are the events "Win" and "Lose" the match, mutually exclusive events. Us the rules of probabilities to justify your answer.