ELEMENTARY STATISTICS MATH 2200
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This 12 page Class Notes was uploaded by Devan Bosco II on Saturday October 3, 2015. The Class Notes belongs to MATH 2200 at Armstrong Atlantic State University taught by Lorrie Hoffman in Fall. Since its upload, it has received 19 views. For similar materials see /class/217865/math-2200-armstrong-atlantic-state-university in Mathematics (M) at Armstrong Atlantic State University.
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Date Created: 10/03/15
Probability Models like Binomial Random Variables and their distributions Administrivia The scoring at bottom left shows 4976 means you have an average so far of 49 from 2 exams and 2 inclass activities and if you earn 90 s the rest of the semester you will have a nal average of 76 DEFN Random Variable is an outcome governed by probability EX What if I roll a die and give you 3 when the 1 comes up and 1 for any other side EEK Q How much do you expect to win ANS Via raw data approach nd the meanaverageuY 1 There are rV models for situations that happen often We will do a simulation create data to arti cially mimic a reallife situation here in class The Binomial Model Number of trials is n a count Each trial is independent Only 2 outcomes Probability of a success on each trial is p a probability We are interested in Xnumber of successes Then the mean of X average number of successes in 11 trials is calculated as np And the variance is nplp The probability of exactly k successes is nC1ltquot pk1p 39k more on this notation later Discovering the Characteristics of a Binomial Model Team Names 1 2 3 4 Q In the real world a coin ip could imitate 5 6 what event Answer Your Experiment was Toss your coin 3 times and record each toss as a Head H or Tai1T and record the number of Heads Each of your team did this 8 times Combine all your results below Describe the outcomes Frequency How many Prob Total number of Heads experiments showed Freq In 3 tosses that number of Heads Total 0 l 2 3 Total 10 Find the Mean Find the Standard Deviation Discovering the Characteristics of 1 a Binomial and of 2 a Geometric Model omit if insuf cient time Team Names 1 2 3 4 Q In the real world a coin ip could imitate 5 6 what event Answer Your Experiment was Toss your coin 6 times and record each toss as a Head H or Tai1T and record the number of Heads Each of your team did this 8 times Combine all your results below Describe the outcomes Frequency How many Prob Total number of Heads experiments showed Freq In 6 tosses that number of Heads Total 0 l 2 3 4 5 6 Total 10 Ch 8 21 a get data off web at httpwwwmatharmstrongedufacultyhoffmanm2200spO6homehtml or type in and use XLregression ageyr as xaxis and price as y Do redyellowblue chart wizard and XYscatter b the older the car the Prices Advertised less you pay decrease is about linear c I see a line so yes d Rsquared is rr which is 945 945 894 14000 12000 10000 To do this by worksheet method rst use XL and ToolsData Analysis Descriptive Statistics to nd means and stdevs nd zscores for year 0 5 10 15 col 3 And price col 4 then multiply those cols for Col 5 add col 5 and diVide its sum by pairs 7 l which is 16 8000 6000 4000 2000 e Rsquared means that 894 of the variability in price can be explained by the year age of the car f some cars will have extra options or will be in better shape and then will command a higher price g XLregession price is Y response and year is X explanatory Rgression Statistics Multiple R 0945 R Square 0894 Adjusted R Square 0887 Standard Error 1220548 Observations 17000 ANOVA df 88 MS F 39 39 39 F Regression 1000 187830720000 187830720000 126083 0000 Residual 15000 22346074118 1489738275 Total 16000 210176794118 f 39 39 Standard Error tStat Pvaue Intercept 12319588 575680 21400 0000 X Variable 1 924000 82289 11229 0000 Click on your XYscattter and then on Chart on the top toolbar and then on add a trendline to see the regression line By hand just pick 2 X values and plug them into your regression line to generate 2 points on the line year PRICE zyr z 1 1 2 3 3 4 4 5 5 6 5990 6 4995 9 3200 9 2250 9 3995 11 2900 11 2995 13 1750 6677559 37362437 000 081 081 081 135 135 189 12995 1351 715998 10950 L351 151762 10495 L081 026222 10995 41811164177 10995 41811164177 6995 41541060537 79900540335068 87000270530964 6995 0270060537 100 121675 4149128 4198654 124866 4176719 106931 10431 138661 product 232 156 111 4194 4194 4103 018 014 4102 000 000 4180 101 4162 145 141 262 1516 4195 ChartTltle g continued price 1231960 7 9240 year h when year0 then pricel231960 we could say this is the new car price ifor every year older the price goes down by 924 ie depreciation jp ce12319604792410 1231960479240307960 k XLregression with year as XaXis and price as yaXis check offboxes to get residuals and residual plots the residuals vs Xage seem to scatter randomly which is good Residuals X Variable 1 Residual Plot 2000 m o o u 5 w n 2000 X Variable 1 But the normal plot needs to be a straight line and looks enough like one meaning that the prices are from a normal distribution which is good Norm al Probability Plot 20000 0 0 50 100 150 Sample Percentile Recall that the way the errors or residuals are found is thusly For the pair 1 12995 first predict nd the point on the regression line by using the formula price 1231960 7 92400 year Here that is 1231960 7 92400 1139560 called yhat or pricehat then the error is the actual price 7 predicted which is 12995 7 1139560 159940 1 ifI plot in 15 4000 and 20 8000 since old corvettes are worth a lot these values make sense but I do not see a good straight line now So I should not do a regression Note that I mixed 2 different sets of data which is usually a bad idea if you are trying to get a good model m ifI plot in 0 35000 must be a corvette then the regression analysis will try to include that odd point and the line will lift to meet it This seems like an outlier and should be investigated n again mixing two very different data sets can be a bad idea Run 2 separate regressions Ch 14 13 a 1 1221113 2 2 1 3 3 1 1 9 4 0 b although don t say with or without replacement let us assume that there are so many MampMs that if we choose 1 out of millions then not putting it back makes next draw 1 out of millions minus 1such a small difference we can ignore it and say it is with replacement 1 PBrn and Em and Brn 3 3 3 027 note conditioning does not matter ie independent 2 Pnot red and not red and red882128 3 888 512 4 1Pnot green and not green and not green 1999271 15 a disjoint b independent c no if disjoint and one event happens then I know so much more about whether the other will happen it can not So they are dependent 24 Iwould have to not lose on all six ie 17 999999153144469 ch 15 3 a 64211768 b 32 c 04 9 a 11 b 111627 c 1127 407 d arage 11112134375 p0 15 assume done without replacement try a tree a 71261142132318 b 1512411310 1 60132012601320955 c 71261151049 84011880071 32 d 5124113102978840950400088 34 PT and LPLlTPT15951425 PF and LPLlFPF65050325 PTlL142514250325 814 Ch 16 7 Money Prob prod expected profit is sum of prod 50000 3 15000 ie 27000 20000 6 12000 wait on 33 and 35 till next test 0 1 0 CI and Testing to compare Means NEW TYPE OF QUESTION Test Whether men at AASU have a mean travel distance that is greater than the mean travel distance for women Use OL 01 Collect data on 3 men and 3 women Ho amen women Ha ummgt women Note these are really too small samples THEORY men Women is atscore when you compare it to a tValue in the ttable A53 your df 2 81211611 ngomen nmen nwomen 2 2 2 2 1 sm 1 nmen 1 nmen nwomen 1 nwomen EEEKKU Let XL do it This number is always between Smallest n 1 and the sum of the n s 2 Let s use the steps I proposed last time lRead the Problem A Ha is lt or gt then it is a onetail test Ha is a not equal to sign then it is a twotail test Note What 0t value is shown in the problem Is the data continuous You Will use the ttables Work with your data COW Find the sample means and look around for s the standard deviation of the data ECalculate df the degrees of eedom by computing n l ie the sample size number of pieces of data and subtract 1 from it OR crazy df if comparing means approximately ltmin of n slgt to ltsum of n s2gt 2 compute a tscore lt u Where u is the guessed mean mentioned in the hypotheses and s was s xH given to you in the problem or for this new situation 3 go to the ttables in A53 ALook for 2tail or onetail B Look for ca so now you know which column C Find the correct row by using your df D Pluck off your tValue EPut a negative in ont of your tValue if there is a lt in your Ha hypothesis 4COMPARE Alf Ha is lt and your tscore lt tValue then go with Ha B If Ha is gt and your tscore gt tValue then go wit Ha C If Ha is not equal to sign then if either A or B is true then go with Ha The ttables are found in text Upper tail probability df 10 05 025 01 1 3078 6314 12706 31821 2 1886 2920 4303 9925 3 1638 2353 3182 4541 45 50 1299 1676 2009 2403 1000 1282 1645 1960 2326 80 90 95 98 Con dence level
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