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WEEK 4 HOMEWORK: LANE CHAPTER 7 AND ILLOWSKY CHAPTERS 6 AND 7

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WEEK 4 HOMEWORK: LANE CHAPTER 7 AND ILLOWSKY CHAPTERS 6 AND 7
THE NORMAL DISTRIBUTION Z-TABLES ARE INCLUDED IN THE COURSE RESOURCES AND YOU ARE TO USE THEM RATHER THAN SOFTWARE TO SOLVE
THESE PROBLEMS. THIS IS STRAIGHT FORWARD TABLE READING.
THIS WEEK’S CONCEPTS ARE REALLY THE HEART OF OUR COURSE. PROBABILITY (FROM LAST WEEK) IS THE DRIVING FORCE BEHIND STATISTICS (AND
THEORETICAL PHYSICS WATCH THE PBS SHOWPARTICLE FEVER”).
THE NORMAL DISTRIBUTION AND THE AREAS UNDER PARTS OF IT ARE ALL WE ARE TRYING TO FIGURE OUT IN STATISTICS. THESE WOULD BE THE
AREAS IN THE TABLE THAT CORRESPOND TO SPECIFIC (CALCULATED) Z-VALUES. OUR DATA ARE THE X-VALUES AND WE CALCULATE OUR Z-VALUES
FROM THEM. WE CAN ALSO BACK-CALCULATE X-VALUES FROM Z-VALUES.
LONG STORY SHORT: THERE ARE SPECIFIC Z-VALUES OF INTEREST REFERRED TO AS “CRITICAL VALUES” AND THOSE ARE THE ONES THAT
CORRESPOND TO SMALL (RARE) AREAS IN ONE OR BOTH “TAILS” OF OUR NORMAL DISTRIBUTION: 1%, 5% OR 10% OF THE TOTAL AREA UNDER THE
NORMAL CURVE. THESE ALREADY SMALL AREAS CAN ALSO BE SPLIT BETWEEN BOTH ENDS LEAVING 0.5%, 2.5% AND 5% IN EACH TAIL. THE
PERCENTAGES ARE OUR “SIGNIFICANCE LEVELS” AND WE CHOSE ONE BASED ON HOW SURE WE WANT TO BE OF OUR CONCLUSION: 90%, 95% OR 99%
(CORRESPONDING TO THE AREAS OF 10%, 5% AND 1%)
NON-TEXT PROBLEM #1: LOOK THESE Z-VALUES UP FOR THESE AREAS: + 1%, + 5%, + 10% AND + 0.5%, + 2.5%% (AND YOU ALREADY HAVE THE + 5%)
AND WRITE THEM DOWN AS THEY DON’T CHANGE. THE Z-VALUES ON THE LEFT END ARE NEGATIVE AND THOSE ON THE RIGHT END ARE POSITIVE.
HERE IS HOW WE USE THESE Z-VALUES TO SEE IF OUR DATA ARE IN THE “RARE” OR “UNUSUAL” AREAS TO THE FAR LEFT (OR RIGHT) OF OUR NORMAL
DISTRIBUTION. WHY DO WE CARE IF DATA ARE RARE? YOU WILL SEE.
LET’S START WITH OUR QUANTITATIVE, CONTINUOUS DATA VALUES (REFERRED TO AS THE X-VALUES). WE NEXT CALCULATE A Z-VALUE FROM OUR X-
VALUE: [Z = (X MEAN) / STD DEV ]. THIS Z-VALUE IS SIMPLY THE NUMBER OF STANDARD DEVIATIONS OUR X-VALUE IS FROM THE MEAN.
NOW, WE COMPARE OUR CALCULATED Z-VALUE (REFERRED TO AS THE “TEST STATISTIC”) TO OUR CRITICAL VALUE (FOR WHATEVER SIGNIFICANCE
LEVEL WE CHOSE).
IF THE TEST STATISTIC IS GREATER THAN THE POSITIVE (+) CRITICAL Z-VALUE WE ARE IN THE “UNUSUAL” OR RARE AREA IN THE RIGHT TAIL OF THE
NORMAL DISTRIBUTION. IF IT IS LESS THAN THE NEGATIVE () CRITICAL VALUE IT ALSO IN THE RARE AREA IN THE LEFT TAIL.
THIS IS THE WAY WE TEST A HYPOTHESIS, AS YOU WILL LEARN IN LATER CHAPTERS. IF OUR TEST STATISTIC ENDS UP IN THE RARE AREA, WE REJECT
OUR HYPOTHESIS, IF NOT WE ACCEPT IT. BUT, KEEP IN MIND THAT “ACCEPTING’ DOES NOT MEAN “PROVING”. STATISTICS PROVES NOTHING BY
ITSELF. IT SIMPLY ADDS SUPPORT TO OTHER INFORMATION.
LET’S GET BACK TO THE NORMAL DISTRIBUTION. THE EXCEL PAGE ATTACHED PROVIDES AN EXAMPLE OF SUCH A DISTRIBUTION. MAKE SURE IT
MAKES SENSE TO YOU.
NOW, LET’S MOVE ON TO THIS WEEK’S HOMEWORK: LANE (C7) FIRST
PROBLEM #8 THIS IS A STRAIGHT FORWARD Z CALCULATION. THIS Z-VALUE (SCORE) TELLS YOU HOW MANY STANDARD DEVIATIONS A SPEED OF 65 IS
FROM THE MEAN (IN THIS CASE IT’S A NEGATIVE DISTANCE, MEANING IT’S BELOW THE MEAN). USE THE CORRECT TABLE (NOT SOFTWARE) TO FIND
THIS AREA (NEGATIVE - Z-VALUES) AND REMEMBER TOO THAT THE TABLES GIVE THE AREAS TO THE LEFT OF THAT Z VALUE. TO GET THE AREAS TO
THE RIGHT (BUT NOT NEEDED IN THIS PROBLEM) YOU SIMPLY SUBTRACT THE AREA TO THE LEFT FROM 1.00, SINCE THE AREA UNDER THE NORMAL
CURVE ACCOUNTS FOR 100% OR 1.00 OF THE DATA.

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KEEP IN MIND THAT THESE “AREAS” ARE THE PERCENTAGE OF DATA BELOW YOUR SPECIFIC X-VALUE AND ALSO THE PROBABILITY (RELATIVE
FREQUENCY) OF DATA BEING IN THAT AREA. THIS IS THE HEART OF IT.
AS AN EXAMPLE, TO READ THE TABLE FOR A Z-SCORE OF +1.31 GO TO THE TABLE OF POSITIVE Z-SCORES AND FIND 1.3 IN THE LEFT COLUMN AND
THEN GO OVER TO THE 0.01 ON THE TOP AND GET THE READING FROM THE TABLE. IT’S 0.9049. IF YOU HAVE A NEGATIVE Z-SCORE USE THE OTHER
TABLE. WHAT IF YOU HAVE 1.315? GET THE AREAS FOR 1.31 AND 1.32 AND IN OUR CASE SIMPLY APPROXIMATE THE VALUE BETWEEN THEM.
BACK TO THE AREA OF 0.9049, THIS AREA MEANS THAT 90.49% OF OUR DATA ARE BELOW THE DATA POINT THAT HAS A CALCULATED Z-VALUE OF
1.31. IT ALSO MEANS THAT THERE IS A 90.49% PROBABILITY THAT A DATA POINT WILL BE IN THAT AREA BELOW OUR DATA VALUE. CONVERSELY,
ABOUT 9.5% WILL BE ABOVE IT (100% - 90.49% = 9.5%). MAKE SENSE?
PART B IN THIS PROBLEM WE USE AN EVEN SMALLER PERCENTAGE (LESS THAN 1%, BUT YOU NEED TO CALCULATE AND FIND THE AREA UNDER THE
CURVE)
PART C IS A LITTLE TRICKIER. NOTE THAT THE PROBLEM STATES THAT ONLY 10% BE OVER THE SPEED LIMIT. OUR TABLES GIVE
PERCENTAGE UNDER A SPECIFIC Z-VALUE. SO, IF 10% ARE OVER, THAN MEANS THAT 90% ARE UNDER. FIND THE Z-VALUE IN THE TABLE (+TABLE)
CORRESPONDING (NEAREST NUMBER) TO 90%. ONCE YOU HAVE THAT Z-VALUE YOU CAN BACK-CALCULATE TO THE ACTUAL X-VALUE, WHICH IS THE
SPEED OF THE CAR. (THE FORMULA IS THE Z = (X-MEAN)/STD DEV SO USE ALGEBRA TO SOLVE FOR “X”)
PART D THE DISTRIBUTION (SHAPE OF THE CURVE) IS ACTUALLY SKEWED NOT BELL SHAPED, TELL ME IF IT’S A POSITIVE OR NEGATIVE SKEW.
PROBLEM #11: ANOTHER TRICKY ONE. THE “TOP 30%” ARE THE ONES TO THE RIGHT SO WE NEED THE Z-VALUE THAT CORRESPONDS TO THE 70% OF
THE AREA BEING BELOW IT. ONCE YOU HAVE FOUND THAT Z-SCORE, BACK-CALCULATE THE ACTUAL X-VALUE, WHICH IS THE NECESSARY SCORE
ASKED FOR. USE THE SAME PROCEDURE FOR PART B (5% ABOVE MEANS 95% BELOW)
PROBLEM #12: PARTS “a” IS FAIRLY STRAIGHT FORWARD, BUT PART “b” IS VERY TEDIOUS. THIS PROBLEM INVOLVES USING THE NORMAL DISTRIBUTION
(THE BELL CURVE ) WHICH IS CONTINUOUS TO APPROXIMATE THE BINOMIAL WHICH IS DISCRETE. REMEMBER THAT A BINOMIAL HAS JUST TWO
OPTIONS (LIKE HEADS/TAILS). THE NORMAL DEALS WITH CONTINUOUS DATA.
WE WANT THE PROBABILITY OF GETTING 15 TO 18 HEADS OUT OF 25 COIN FLIPS. IF YOU LOOK AT THE HISTOGRAM THAT HAS THE NORMAL BELL SHAPE
YOU SEE THAT EACH COLUMN IS CENTERED ON THE 1, 2, 3 ETC. THIS MEANS THAT FOR A VALUE OF 2 THE COLUMN COVERS THE RANGE OF 1.5 TO
2.5. DO YOU SEE THIS? OK SO HERE IS WHAT WE DO TO USE THE NORMAL TO APPROXIMATE THE BINOMIAL
FIRST: WE NEED TO CALCULATE THE MEAN, WHICH HERE EQUALS THE TOTAL NUMBER OF DATA POINTS (IN THIS CASE 25 TOSSES) TIMES THE
PROBABILITY OF GETTING A PARTICULAR RESULT (LIKE A HEAD), IN THIS CASE 50/50 OR 0.5. THE VARIANCE EQUALS THE TOTAL NUMBER OF DATA
POINTS TIMES THE FIRST PROBABILITY (GETTING HEADS), WHICH IS 0.5 TIMES THE PROBABILITY OF NOT GETTING HEADS (TAILS IS ALSO 0.5) YOU CAN
SEE THAT WE COULD HAVE DIFFERENT PROBABILITIES, FOR EXAMPLE IF THE COIN WERE WEIGHTED SO IT FAVORED HEADS, BUT THAT IS NOT THE
CASE HERE. ONCE WE CALCULATE THIS VARIANCE, THE STANDARD DEVIATION IS SIMPLY THE SQUARE ROOT OF THE VARIANCE. WE NOW HAVE OUR
SAMPLE MEAN AND STANDARD DEVIATION FOR OUR BINOMIAL.
TO GET PROBABILITIES FROM OUR TABLES WE NEED Z-VALUES. WE WANT THE PROBABILITY OF GETTING 15 TO 18 HEADS OUT OF 25 TOSSES. BUT,
SINCE THE NORMAL DISTRIBUTION IS CONTINUOUS, THE 15 GOES FROM 14.5 TO 15.5 AND 18 GOES FROM 17.5 TO 18.5. SO, OUR RANGE WHEN USING
THE NORMAL TO APPROXIMATE THIS BINOMIAL IS 14.5 UP TO ____? NOW, CALCULATE THE Z-VALUES FOR THESE TWO LIMITS.
FROM THESE Z-VALUE WE GO TO THE TABLE AND FIND THE AREAS TO THE LEFT CORRESPONDING TO THEM. SINCE WE ONLY WANT THE AREA
BETWEEN THESE TWO LIMITS WE MUST SUBTRACT THE LOWER AREA FROM THE UPPER AREA. IS THIS CLEAR TO YOU? LOOK AT TEXT PICTURES TO
SEE WHAT IS GOING ON.
PART B: INVOLVES THE ACTUAL BINOMIAL EQUATION. THE BINOMIAL GRAPH IS NOT CONTINUOUS BUT WILL HAVE A COLUMN FOR EACH POSSIBILITY,
IN THIS CASE 15, 16, 17, AND 18 HEADS. WE NEED TO USE THE COMPLEX BINOMIAL FORMULA THAT WE WORKED OUT IN WEEK 3 FOR PROBLEM LANE
CHAPTER 5 #25. CHECK IT OUT.

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Wk4-HW4 Instructions WEEK 4 HOMEWORK: LANE CHAPTER 7 AND ILLOWSKY CHAPTERS 6 AND 7 THE NORMAL DISTRIBUTION Z-TABLES ARE INCLUDED IN THE COURSE RESOURCES AND YOU ARE TO USE THEM RATHE THESE PROBLEMS. THIS IS STRAIGHT FORWARD TABLE READING. THIS WEEK’S CONCEPTS ARE REALLY THE HEART OF OUR COURSE. PROBABILITY (FROM LAST WEEK) IS THE DRIVING F THEORETICAL PHYSICS – WATCH THE PBS SHOW “PARTICLE FEVER”). THE NORMAL DISTRIBUTION AND THE AREAS UNDER PARTS OF IT ARE ALL WE ARE TRYING TO FIGURE OUT IN STATISTIC AREAS IN THE TABLE THAT CORRESPOND TO SPECIFIC (CALCULATED) Z-VALUES. OUR DATA ARE THE X-VALUES AND W FROM THEM. WE CAN ALSO BACK-CALCULATE X-VALUES FROM Z-VALUES. LONG STORY SHORT: THERE ARE SPECIFIC Z-VALUES OF INTEREST REFERRED TO AS “CRITICAL VALUES” AND THOSE A CORRESPOND TO SMALL (RARE) AREAS IN ONE OR BOTH “TAILS” OF OUR NORMAL DISTRIBUTION: 1%, 5% OR 10% OF TH NORMAL CURVE. THESE ALREADY SMALL AREAS CAN ALSO BE SPLIT BETWEEN BOTH ENDS LEAVING 0.5%, 2.5% AND 5% PERCENTAGES ARE OUR “SIGNIFICANCE LEVELS” AND WE CHOSE ONE BASED ON HOW SURE WE WANT TO BE OF OUR C (CORRESPONDING TO THE AREAS OF 10%, 5% AND 1%) NON-TEXT PROBLEM #1: LOOK THESE Z-VALUES UP FOR THESE AREAS: + 1%, + 5%, + 10% AND + 0.5%, + 2.5%% (AND Y AND WRITE THEM DOWN AS THEY DON’T CHANGE. THE Z-VALUES ON THE LEFT END ARE NEGATIVE AND THOSE ON THE HERE IS HOW WE USE THESE Z-VALUES TO SEE IF OUR DATA ARE IN THE “RARE” OR “UNUSUAL” AREAS TO THE FAR LEFT DISTRIBUTION. WHY DO WE ...
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