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2.2 Normal Distributions Homework Answers 41. Men’s Height (inches) 41. Weight (ounces) 43. a) b) c) d) Approximately 2.5% Between 64 and 74 inches Approximately 13.5% 84th percentile 44. a) b) c) d) Approximately 2.5% Between 9.07 and 9.17 ounces Approximately 83.85% 16th percentile 45. The standard deviation is approximately .2 for the taller distribution, and .5 for the shorter one 46. The mean is 10 and the standard deviation is about 2 47. a) b) c) d) .9978 .0022 .9515 .9493 48. a) b) c) d) .0069 .0069 .1798 .1004 49. a) b) .8587 .2718 50. a) b) .7621 .241 51. a) b) Z= -1.28 Z= .41 Assessing Normality Some distributions are normal distributed (or approximately normally distributed) SAT scores Gas mileage Chapter 1 test scores Other variables are not normally distributed Household income Military spending Survival time after cancer diagnosis Etc. Assessing Normality So, before we use techniques that work for normal distributions, we need to have a way (or ways) to assess whether a distribution can be considered a normal distribution We will use several methods to assess whether a distribution is normally distributed Look at the distribution 2. 68-95-99.7 rule 3. Normal probability plot 1. Look at the distribution Is this normally distributed? Look at the distribution Is this normally distributed? Probably not. Looks more like a uniform distribution Look at the distribution When we look at a distribution to see if it is normally distributed, what are we looking for? Look at the distribution When we look at a distribution to see if it is normally distributed, what are we looking for? Symmetric Bell-shaped Single-peaked So what about this one? Is it (more or less) symmetric? Is it bell shaped? Is it single-peaked? So what about this one? Is it (more or less) symmetric? Debatable, but I would say yes Is it bell shaped? NO Is it single-peaked? NO And this one? Is it (more or less) symmetric? Is it bell shaped? Is it single-peaked? And this one? Is it (more or less) symmetric? YES Is it bell shaped? More like 2 bells Is it single-peaked? NO And this one? Is it (more or less) symmetric? NO Is it bell shaped? YES Is it single-peaked? YES 68-95-99.7 Rule We can use the 68-95-99.7 rule to assess normality as well In addition to its other uses in estimating percent to the left/right We know that in ANY true normal distribution, 68 percent of the observations will fall within 1 standard deviation of the mean Remember, no real-world distribution is PERFECTLY normal But some are close enough approximations, while others are not So what we are really doing is assessing whether it is close enough to think of it as being normal 68-95-99.7 Rule Take the data to the right 29 12.7 How would we go about using 68-95- 99.7 rule? What would be the first step? 26.4 19.4 18.8 21.6 17 10.3 23.7 26.6 68-95-99.7 Rule Take the data to the right 29 12.7 How would we go about using 68-95-99.7 rule? What would be the first step? Calculate the mean The mean here is 20.6 What would we do next? 26.4 19.4 18.8 21.6 17 10.3 23.7 26.6 68-95-99.7 Rule Take the data to the right 29 12.7 How would we go about using 68-95-99.7 rule? What would be the first step? Calculate the mean The mean here is 20.6 What would we do next? Calculate the standard deviation Here it is 6.1 Then what do we need to do? 26.4 19.4 18.8 21.6 17 10.3 23.7 26.6 68-95-99.7 Rule How would we go about using 68-95-99.7 rule? What would be the first step? Calculate the mean The mean here is 20.6 What would we do next? Calculate the standard deviation Here it is 6.1 Then what do we need to do? Calculate Z-scores for each observation (see right) Now what? 1.38 -1.28 0.95 -0.19 -0.29 0.17 -0.58 -1.67 0.51 0.99 68-95-99.7 Rule Calculate Z-scores for each observation (see right) Now what? Now we compare to the 68-95-99.7 expectation Are there about 68% between 0 and 1 standard deviations from the mean? Yes, 7 out of 10 have Z-scores between -1 and 1 Are there about 95% between -2 and 2? Yes, 10 out of 10 Are there about 99.7% between -3 and 3? Yes, 10 out of 10 This provides us an indication that it is probably normally distributed 1.38 -1.28 0.95 -0.19 -0.29 0.17 -0.58 -1.67 0.51 0.99 Normal Probability Plots Which of these do you think is ‘normal?’ Normal Probability Plots This one Not this one Normal Probability Plots Graph the actual values versus the z-scores A completely normal distribution will be a perfectly straight line When assessing normality, we will never see something perfectly straight—if it is close, then we can consider it normal Normal Probability Plots Here is the normal probability plot of our 10-observation dataset from earlier. We concluded from the 68-95-99.7 rule that it was normally distributed. Does it look normal here too? 2 1.5 1 0.5 0 0 -0.5 -1 -1.5 -2 5 10 15 20 25 30 35 Let’s Try It Make a normal probability plot of the following data: 9, 1, 5, 3, 8, 1 Should we consider this to be normally distributed? Let’s Try It Should look like this: 1.5 1 0.5 0 0 -0.5 -1 -1.5 2 4 6 8 10 Technology Corner (Page 128) Click the STAT button Enter your data under L1 Click 2ND then STAT PLOT (y= button) Plot 1 will be highlighted Hit enter Turn it ON Choose option 6 for type Data list should be L1 Data axis should be X Hit graph Assessing Normality Which of our three methods do you prefer? Assessing Normality Look at it Downside: imprecise—some wiggle room Using 68-95-99.7 rule Downside: more effective with bigger datasets Normal Probability Plots Downside: most time-intensive But also the most precise Normal Probability Plots on the AP Exam They will NEVER specifically ask you to make one They will NEVER specifically ask you to interpret one BUT, they CAN ask you to assess the normality of a certain dataset You could do this using the 68-95-99.7 rule But you could also do this by making a normal probability plot Whatever makes the most sense to you Chapter 1 Test (Raw Scores) Chapter 1 Test Raw Score Statistics Mean: 64.85 Median: 65 Standard Deviation: 11.24 Q1: 60.5 Q3: 71.75 Chapter 1 Test Curve This test was curved Not scaled! First, I took the z-score for each score (value minus mean) divided by standard deviation I then decided what I wanted the distribution to look like I set it up so that the mean was 80, and the highest score was a 100 This meant using a standard deviation of approximately 8.28 Using these numbers, I calculated your curved score by adding your (Z- score times 8.28) to 80. So if your Z-score was exactly 1, you would have a score of 88.28. Curved Scores Then add extra Credit This is what shows up in Infinite Campus Take the curved score, and add your extra credit value if you did the summer homework
