Normal distribution slide share
- 1. Normal Distribution Mock test Friday Week 2! / Sample Assessments
- 2. God’s Mean This used to be called the ‘error curve’ – how far (say) people’s heights deviated from ‘God’s mean’. We know now that data occurring ‘naturally’ clusters about the mean, but spreads out more and more thinly towards the extreme values. Pear orchard example Miss Fuller made in “ Basketball player? God’s image”
- 3. Standard Deviation If you don’t have a clue what SD really is, try this: Measure your thumbs The SD is a measure of how far away from the mean your scores are. Our thumb ‘mean’ is 61.79 mm. On average, our 18 scores were 4.86 mm away from the mean, so our SD is 4.86 mm. In the diagram below, 68% of the thumbs will be 1 SD away from the mean, 95% of our thumbs will be 2 SDs away from the mean, and 99.7% will be 3 SDs away from the mean. Let’s see how close we get:
- 4. Standard Deviation How many ‘thumbs’ will we expect in each slice? Put yours in! 47.20 52.07 56.93 61.79 66.65 71.51 76.38 68% of 19 is 12.969, 95.44% of 19 is 18.134 and 99.7% of 19 is 18.95 Even with such a small sample, the standard deviation probs work!
- 5. Textbook references - Theta Normal Distribution Curves Ex 24:01 1 – oral Standard Deviation Ex 24:02 1 – read p 405 Q4 – scan gymnasts and disc Normally Distributed data Ex 24:03 1 – DOIT. Need words. Tables – p 418 dim. TC or GC The Standard Normal Distribution Ex 24:04 1 TC Converting to the standard normal Ex 24:05 (and 06?) 2 use brains? Contextual Standard Normal Distribution 24:07 24:08 2 Inverse Normal Distribution Ex 24:09 1 Applications using the ‘Inverse’ Normal z Ex 24:11 value 2 Practice Assessment Work 8
- 6. The Normal Distribution- definitely!! 34% 0.5% 0.5% 2% 13.5% 13.5% 2% 34% -3 -2 -1 mean +1 SD +2 +3
- 7. We now know that naturally occurring data falls like this: 68.26% of data falls within 1 sd either side of the mean. It is _________ or _________ that data falls in this region. 95.44% of data falls within 2 sd either side of the mean. It is ___________ or ____________ that data falls in this region. Green Thetad 30.2 p.353 99.74% of data falls within 3 sd either side of the mean. It is _________ _____________ that data falls in this region. If you become very familiar with these numbers you can estimate your answers – a great ‘mental check’!
- 8. We now know that naturally occurring data falls like this: 68.26% of data falls within 1 sd either side of the mean. It is _________ or _________ that data falls in this region. 95.44% of data falls within 2 sd either side of the mean. It is ___________ or ____________ that data falls in this region. Green Thetad 30.2 p.353 99.74% of data falls within 3 sd either side of the mean. It is _________ _____________ that data falls in this region. If you become very familiar with these numbers you can estimate your answers – a great ‘mental check’!
- 10. Just in case…
- 11. Standard Normal Distribution This is a perfect world called Zed The mean μ is always 0 and the standard deviation σ is always 1 We have tables to calculate the probability that (say) a value is 1.5 or greater, or between 0 and 2. The curve is called ‘the normal curve’ and the total area underneath it is ONE. The area under each half is 0.5. I will refer to this as ‘The Z World’ Have a look at these examples:
- 12. The Normal Distribution- definitely!! 34% 0.5% 0.5% 2% 13.5% 13.5% 2% 34% -3 -2 -1 mean +1 SD +2 +3
- 13. Practice Find the probability that x is between… 1. 0 and 1.55 2. 0 and 0.54 3. 0 and 0.9 4. 0 and 0.04 5. 0 and 1.57 6. -1.4 and 0 7. -1.20 and 0 8. -1.3 and 1.3 9. 0.72 and 1.8 10. 1.8 and -0.05 Discuss differences column!!
- 14. Practice – Answers! Find the probability that x is between… 1. 0 and 1.55 2. 0 and 0.54 3. 0 and 0.9 4. 0 and 0.04 5. 0 and 1.57 6. -1.4 and 0 7. -1.20 and 0 8. -1.3 and 1.3 9. 0.72 and 1.8 10. 1.8 and -0.05 1. 0.4394 2. 0.2054 3. 0.3159 4. 0.0160 5. 0.4418 6. 0.4192 7. 0.3849 8. 2x0.4032 = 0.8064 9. 0.4641 - 0. 2642 10.0.4641 + 0.0199
- 15. Harder Practice Find the probability that x is between… 1. 0 and 0.231 2. 0 and 1.03 3. Find P( Z>2.135) 4. Find P( Z>2.135) 5. Find P( Z>-0.596) 6. Find P( Z>2.135) 7. Find P( Z<0.582) 8. Find P( Z<-1.452) Sorry - no answers!
- 16. Mapping onto the ‘REAL’ World 0 1.2 We can see that if Z was 1.2, the probability of a point lying between 0 and 1.2 is… That is, 38.49% or 0.3849 of our sample lies between 0 and 1.2 Can you see our problem when it comes to ‘Real Life’?!! This information in our tables is USELESS for data which is doesn’t have a mean of 0 and a sd of 1 UNLESS… we can MAP our real life data onto the standard normal distribution. I will call this ‘the Real Life X world’
- 17. Mapping - step by step Imagine this graph shows the X world, where the mean (μ) is 5 and the SD(σ) is 2. Stripes show 1 standard deviation. -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 BUT in our maths Z world the mean is 0 and the SD is always 1. What must happen to map the graph above onto the Z world? The graph moves DOWN 5 to make the mean 0 and then compresses so the SD is now 1. -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
- 18. How do we map the X world onto the Z? To go from X to Z (so we can use our tables) we... take off (subtract) the mean and divide by the standard deviation. And that gives us our formula: Z= 푋 − μ σ Z(scor푒) = (푋푠푐표푟푒) −푚푒푎푛 푠푡푑 푑푒푣′푛
- 19. From start to finish… The heights of BC students are normally distributed, with a mean height of 1.7m and a standard deviation of 0.1 m. What percentage of students will be taller than 1.87m? Of the 700 students, how many will be shorter than 1.87m? 1. ALWAYS draw a diagram. Put info in and shade what you need. 2. Estimate the answer and write it in. 3. This data is in ‘the X world’. Use the formula (next slide) to translate it into the Z world (mean 0, sd = 1).
- 20. The Maths 4. Translate your X world to the Z world. 5. Now we can use the tables and look up 1.7. Z= 푋 − μ σ P(X>1.87) = P(Z>1.87-1.7) 0.1 = P(Z>1.7)
- 21. The Answer 6. The answer from the tables is 0.4554. 7. Look at your diagram and adjust for your answer. 0.5 – 0.4554 = 0.0446 8. Look back at the original estimate – how close? 9. Read the question and give your real life answer. P(X>1.87) = P(Z>1.87-1.7) Z= 푋 − μ σ 0.1 = P(Z>1.7) Prob that z is between 0 and 1.7 is… 0.4554 10. The probability that a student is taller than 1.87m is 4.46% 11. Have we finished? – nooo!
- 22. For how many questions, we multiply the probability x the number. In this case we want to know how many are SHORTER than 1.87m. Our tables gave us 0.4554, what else must we do? Add 0.5 So 0.9554 x 700 = 668.78 Give the maths and then answer the question with a ‘real life’ answer. There are about 669 students that are shorter than 1.87m . Try the classic Christmas tree problem, next. ‘How many’ questions
- 23. The Christmas Tree Problem The lengths of Christmas tree branches from a pine-tree plantation can be assumed to be normally distributed with a mean length of 1.8m and a standard deviation of 20 cm. What percentage of branches would measure less than 2.07 m? If there were 80 branches, how many would we expect to be less than 2.07 m in length?
- 24. Basics: Step 1: Write down important information. μ= 1.8 m sd(σ) = 0.2 m P(X< 2.07) = ? Step 2: Convert to Z score (sub mean, ÷ by sd): P( Z < 2.07 – 1.8 ) 0.2 Step 3: ALWAYS draw a diagram – Doit!
- 25. Step 4: Estimate an answer… We already have an idea of a ‘good’ answer now: more than 0.5 + 0.34 = 0.84 About 0.84 or more is good
- 26. Step 5: Look up Z in the tables: Z = 1.35 gives us 0.4115 Step 6: Adapt as required and answer the question: P(Z<1.35) = 0.5 + 0.4115 = 0.9115 Step 7: Real life Answer - The probability that branches measure less than 2.07 m is 0. If there were 80 branches, how many would we expect to be less than 2.07 m in length? How many problems: _______ x _______ = ________ Answer…?? Round for real life!
- 27. Workbook – READ pages 34 and 35, or better still, cover the model answers and do each step, uncovering and checking as you go Do p 37 – 42 Graphics Calculator ppl – read p 36. Sky tower theta – READ p 341 Exercise 24.02 p 342 (10 mins only) then… Ex 24.03 p 348 Ex 24.04 p 349 and 24.06 (the best and hardest) p 352 Expected value – 24.07 p354 Problems to try… Green theta – READ p 341 Exercise 30.2 p 353 (10 mins only) then… Ex 30.3 p 358 Ex 30.4 p 359 (the best and hardest – this combines expected value)
- 28. Inverse - essential steps 1. ALWAYS draw a diagram! 2. After you have done that, draw one for the ‘Z’ world. The probabilities are the same AND the tables ‘work’. 3. To find your missing z point, look up the prob in the BODY of the tables. – GO LOW. 3. Adjust your answer for the diagram – negative? 4. Then use the force-errr formula to get your x point. 5. Write your ‘real life’ answer with appropriate rounding.
- 29. Mazda Man’s lightbulbs Light bulbs last for 200 hours on average, with a standard deviation of 40 hours and they are normally distributed. The Mazda Man wants to create an ad that promises that 80% of bulbs last for at least a certain number of hours. He is asking you to work out the hours. Real world diagram! Z world diagram. Look up 0.3 in the BODY of the tables – how close can you get? 0.2995 – we still need 5 more – differences – how close can you get? 3 – look up and 1 is your last digit. The z number is… 0.841 Adjust for your diagram – positive or negative?
- 30. Mazda Light bulbs ctd Now do the maths In the diagram above we HAVE z = -0.841 and we need x. -0.841 = x – 200 40 Multiply by 40 and add 200. X = 166.36 Do a ‘sense check’ with your problem – does this sound right? Now apply to real life.. 80% of bulbs will last longer than 166.36 hours (yuk!) 80% of bulbs will last longer than 166 hours and 21.36 minutes (yuk!) I would advise the Mazda man… Z= 푋 − μ σ
- 31. Lambsie-pies Newborn lamb weights are normally distributed about 1.5 kg with a standard deviation of125 grams. What birth weight is exceeded by 30% of newborns? Real world diagram! Z world diagram. Look up 0.2 in the BODY of the tables – how close can you get? 0.1985 – we still need 15 more – differences – how close can you get? 14 – look up and 4 is your last digit. The z number is… 0.524 Adjust for your diagram – positive or negative?
- 32. Mazda Light bulbs ctd Now do the maths In the diagram above we HAVE z = 0.524 and we need x. 0.524 = x – 1.5 0.125 X = 1.57 kg Z= 푋 − μ σ Do a ‘sense check’ with your problem – does this sound right? Now apply to real life.. I expect that 30% of the lambs will exceed the birth weight of 1.57 kg
- 33. Inverse problems Green Theta p 366 An army recruiting officer measures the length of the feet of all new recruits before outfitting them with Army-issue boots. He knows that these foot lengths are normally distributed with a mean of 260mm and a SD of 15 mm. 12% of the recruits are like Jack Reacher and have feet so large that they do not fit any of the boots. Find the maximum foot length to the nearest mm which the army issue boots fit. Workbook p 46 and 47
- 34. 2 4 6 8… you’re in the army now, son. Real world diagram! Z world diagram. Percentage?? Look up ____ in the BODY of the tables – how close can you get? The z number is… _____ Adjust for your diagram – positive or negative? Now do the maths In the Z diagram we HAVE z = _____ and we need x. 0._____ = x –….. X = ________