2. 1.Assemble the following tools: Graphing calculator z-tables (modules 4 - 5)z-tables Paper and pencil Reference for calculator keystrokes 2.Complete the diagnostic to assess your prior knowledge or at least review the diagnostic to help you frame the content within this module. Before beginning this module

3. 1.Access solutions.solutions 2.If you did not score 100% or did not take the diagnostic, continue with this module 3.If you score 100%, you may proceed to the next module. Next Steps

4. Standardized values ( z -scores) have no units. A negative z -score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean. z -scores measure the distance of each data value from the mean in standard deviations. A z -score of 1.4 would tell us the data value is 1.4 standard deviations above the mean. Previously Learned: z -score

5. Outcomes Identify the properties of a normal probability distribution. Describe how the standard deviation and the mean affect the graph of the normal distribution, Given a normally distributed data set, apply the empirical rule to approximate a probability or a percentile.

6. Length of Right Foot Number of People with that Shoe Size 8765432187654321 4 5 6 7 8 9 10 11 12 13 14 Suppose we measured the right foot length of 30 teachers and graphed the results. Assume the first person had a 10 inch foot. We could create a bar graph and plot that person on the graph. If our second subject had a 9 inch foot, we would add her to the graph. As we continued to plot foot lengths, a pattern would begin to emerge. Adapted from Understanding the Normal Curve presentation by Del Siegle, University of Conneticut.

7. Length of Right Foot 8765432187654321 If we were to connect the top of each bar, we would create a frequency polygon. Notice how there are more people with a 10 inch right foot than any other length. Notice also how as the length becomes larger or smaller, there are fewer and fewer people with that measurement. This is a characteristics of many variables that we measure. There is a tendency to have most measurements in the middle, and fewer as we approach the high and low extremes. Number of People with that Shoe Size 4 5 6 7 8 9 10 11 12 13 14 Adapted from Understanding the Normal Curve presentation by Del Siegle

8. You will notice that if we smooth the lines, our data almost creates a bell-shaped curve. This mound/bell-shaped curve is sometimes referenced as the “Bell Curve” or the “Normal Curve.” Length of Right Foot Number of People with that Shoe Size 8765432187654321 4 5 6 7 8 9 10 11 12 13 14 The two skinny parts where the curve approaches the x- axis are the tails. Adapted from Understanding the Normal Curve presentation by Del Siegle

9. Distribution Length of Most Popular Boy Names in 2009 Quantitatively we can calculate some descriptive statistics. n = 500 names µ = 5.7 letters σ= 1.3 letters Mode is 6 letters Median is 6 letters Number of Letters Number of Babies What if we wanted to determine the probability of a selecting a name 6 letters long? Previously, we were studying how to describe a data set. We can see that this data is mound-shaped, unimodal, and somewhat symmetrical. Adapted from Making Sense of Change: New Statistic Standards in Algebra by Patrick Lintner, VCTM Fall Conference

10. What is the probability of selecting a name between 1 and 13 letters? Number of Letters Number of Babies Length of Most Popular Boy Names in 2009 To find the probability based on the given data, we need to know the relative frequency: 1 10 71 137 153 89 26 9 2 2 All the relative frequencies add up to 1. Subtract the events that are not included in the problem from 1: What is the probability of selecting a name greater than or equal to 3 letters, but less than or equal to 9 letters? Based on the data collected, what is the probability, that a popular name selected will be greater than 11? Of course, we could add up each of the relative frequencies: Since it includes all the outcomes then it is or 1 Since there are no values greater than 11, then the probability is 0. Notice how the relative frequency will be directly related to the bar height or the relative area of each bar in comparison to others. What if we wanted to determine the probability of a selecting a name 6 letters long? Adapted from Making Sense of Change: New Statistic Standards in Algebra by Patrick Lintner, VCTM Fall Conference

11. The mean,mode,and median 12 13 14 15 16 17 18 19 20 21 22 Points on a Quiz Number of Students 987654321987654321 12+13+13+14+14+14+14+15+15+15+15+15+15+16+16+16+16+16+16+16+16+ 17+17+17+17+17+17+17+17+17+18+18+18+18+18+18+18+18+19+19+19+19+ 19+ 19+20+20+20+20+ 21+21+22 = 867 867 / 51 = 17 12 13 14 14 15 15 15 16 16 16 16 17 17 17 17 17 17 17 17 17 18 18 18 18 19 19 19 20 20 21 22 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22 will all fall on the same value in a normal distribution. Now lets look at quiz scores for 51 students. Adapted from Understanding the Normal Curve presentation by Del Siegle

12. Normal distributions (bell shaped) are a family of distributions that have the same general shape. 1.The mean, median and mode are equal. 2.The graph of a normal distribution is called a NORMAL CURVE. 3. A normal curve is single-mound (bell shaped) and symmetrical about the mean. 4. A normal curve never touches, but gets closer and closer to the x-axis (approaching f(x) = 0) as it gets farther from the mean.

13. Examples of data that approximate the Normal Model The measure of LDL cholesterol in adults Heights of each gender IQ Scores SAT and ACT scores The width of stripes on a zebra Rainfall acidity in the Shenadoah Mountains Lifetime of products The size of eggs of young hens

14. The height (ordinate) of a normal curve is defined as: where  is the mean and  is the standard deviation,  is the constant 3.14159, and e is the base of natural logarithms and is equal to 2.718282. Mathematical Formula for Height of a Normal Curve Building Knowledge for Teachers x can take on any value from - infinity to +infinity. f(x) is very close to 0 if x is more than three standard deviations from the mean (less than -3 or greater than +3). 0 -3σ+3σ0

15. The mean and standard deviation are useful ways to describe a set of scores. If the scores are grouped closely together, they will have a smaller standard deviation than if they are spread farther apart. Small Standard Deviation Large Standard Deviation Different Means Different Standard Deviations Different Means Same Standard Deviations Same Means Different Standard Deviations Adapted from Understanding the Normal Curve presentation by Del Siegle

16. Match the sketch with the correct mean and standard deviation. A C B 1.μ=12 σ=3.67 2.μ=12 σ= 4 3.μ=6 σ=1.3 4.μ=8 σ=2 D 0 20 0 0 0 ADBCADBC #1 and #2 have the same mean which is true about graphs A and D. Graph D is more spread out so, so it has the larger standard deviation. #3 and #4 have slightly different means. Compared to C, graph B is shifted to left of C indicating a smaller mean. Graph B is also “skinnier” than graph C showing a smaller standard deviation or more tightly clustered data set.

17. Comparing Means, Medians, and Modes Mean Notice the blue, red, and green lines on the distribution graphs below. The blue line represents the mean, the red line represents the median, and the green line represents the mode. This demonstrates how whenever data becomes skewed the mean is affected the most. The bottom graph shows how the mean, median, and mode are about the same on a normal distribution. Left Skewed Right Skewed Normal Distribution Medians Mean Mean, Median and Mode are the Same Modes Adapted from Understanding the Normal Curve presentation by Del Siegle

18. BAC EDF HG I Which of the following graphs represents a normal distributions? Tell why or why not.

19. BAC E D F H G I A. No. There is no symmetry. Also, notice the two peaks instead of single mound. B. No. This graph is skewed left and lacks symmetry. C. Yes. This graph is fairly symmetric and bell-shaped, while not perfect. G. No. There is no symmetry. H. No. This graph is skewed right and lacks symmetry. I. Yes. This graph is fairly symmetric and bell-shaped. D. No. This graph is skewed right and lacks symmetry. E. Yes. This graph is symmetric and bell-shaped. F. No. This is not a single-mound, bell- shaped graph.

20. The Empirical Rule (also known as the 68-95-99.7 rule) The Empirical Rule gives some general statements relating the mean and the standard deviation of a bell-shaped distribution. It relates the mean to one, two, and three standard deviations

21. In a normal distribution about 68% of the values fall within one standard deviation of the mean. µ-1σ µ µ+1σ 68% Since the normal curve is symmetrical, that leaves approximately 16% not shaded in each tail. Since the normal curve is symmetrical, that leaves approximately 16% not shaded in each tail.

22. µ - 2σ µ µ +2σ 95 % In a normal distribution about 95% of the values fall within two standard deviations of the mean. Approximately 5% remains in the two tails. So, approximately 2.5% remains in each tail.

23. µ - 3σ µ µ + 3σ 99.7 % In a normal distribution about 99.7% of the values fall within three standard deviations of the mean. Approximately 0.3% (1/333) of the values are outside three standard deviations from the mean.

24. The Empirical Rule (68-95-99.7 rule) Generally, a certain percentage of the data lies within 1, 2 or 3 standard deviations of the mean

25. z-score -3 -2 -1 0 1 2 3 IQ-score 65 70 85 100 115 130 145 SAT-score 200 300 400 500 600 700 800 The number of values that one standard deviations equals varies from distribution to distribution. 0.15% 2.35% 13.5% 34% 2.35% 0.15% -3σ -2σ -1σ 0 1σ 2σ 3σ IQ Score: µ=100 σ=15 SAT Score: µ=500 σ=100 Notice the percentage approximations are symmetric about the mean. Adapted from Understanding the Normal Curve presentation by Del Siegle

26. On one math test, a standard deviation may be 7 points. If the mean were 45, then we would know that 68% of the students scored from 38 to 52. On another test, a standard deviation may equal 5 points. If the mean were 45, then 68% of the students would score from 40 to 50 points. 24 31 38 45 52 59 63 Points on Math Test 0.15% 2.35% 13.5% 34% 2.35% 0.15% -3σ -2σ -1σ 0 1σ 2σ 3σ 30 35 40 45 50 55 60 Points on a Different Test 0.15% 2.35% 13.5% 34% 2.35% 0.15% -3σ -2σ -1σ 0 1σ 2σ 3σ

27. Many variables in real world have bell-shaped curves This made the bell-shape curve or Normal Curve a useful probability distribution. Data can be approximately “normal” which is good enough for making certain statistical predictions.

28. By continuous, we mean the amount of toxic spray could be 61.757 ounces, then 61.7578 ounces, etc. So, with continuous random variables, we cannot determine the probability of a specific value because there are an infinite number of possible values. You may have noticed that we are no longer using histograms to present data. We are using a smooth curve instead. When we apply the normal model we are addressing very large data sets of continuous data. By very large data sets, we mean all boys ages 10-12 versus a class of 100 boys.

29. This is why with continuous random variables, we only calculate the probabilities of certain intervals. Intervals such as … Finding the probability that a Harry Homeowner used less than 67.5 ounces of spray; Or the probability of using between 58 and 67 ounces of spray per year. Asking whether the probability of the amount of toxic being sprayed is less than versus less than or equal is not significant since the difference is so small.

30. 6 12 18 24 30 36 40 Fuel Economy (mpg) -3σ -2σ -1σ 0 1σ 2σ 3σ Try this problem Suppose the fuel economy for cars in the Manassas area is distributed approximately normal with a mean of 24 mpg and a standard deviation of 6 mpg. What percent of the cars driven in the Manassas Area get between 12 and 36 mpg? 1.Sketch a normal model, marking the mean and 3 standard deviations in each direction. 2.Shade the area of interest. The data lie within 2 standard deviations of the mean. Since the desired values are exact standard deviations from the mean, we can use the empirical rule (68-95-99.7) to get an approximate percent. Approximately 95% of the cars driven in Manassas get between 12 and 36 mpg. 95%

31. Another Problem to Try The heights of 3rd grade boys is normally distributed with a µ=50 inches and σ=5inches. What percent of 3rd grade boys have heights between 45 and 55 inches? (A sketch helps students to visual the problem) 45 and 55 are each 1 standard deviation from the mean. The Empirical Rule states that approximately 68% of the data lie within 1 standard deviation of the mean. Approximately 68% of the third grade boys have heights between 45 and 55 inches. 35 40 45 50 55 60 65 Height (inches) -3σ -2σ -1σ 0 1σ 2σ 3σ 68%

32. More Practice Scores on the IQ-Test have an approximately normal distribution. The mean score is 100 and the standard deviation is 15. What percent of the population has a score less than 115? We can use the symmetry of the normal curve to help us solve this. Left of the mean is 50%. 1 standard deviation to the right of the mean is half of 68% or 34%. 84% of the population scores below 115. What percent of the population has a score greater than 115% Now the problem is asking for the non-shaded area, 100 – 84 = 16. 16% of the population scores higher than 115 -3σ -2σ -1σ 0 1σ 2σ 3σ 55 70 85 100 115 130 145 IQ Score 50% 34% You can also use the symmetry of the normal curve in another way. 115 is 1 standard deviation from the mean. The 1 standard deviation from the mean is 68%. Therefore, the total in both tails is 32% (100-68) and in each tail is 16%. So 68 + 16 = 84% 55 70 85 100 115 130 145 IQ Score -3σ -2σ -1σ 0 1σ 2σ 3σ 16% 68%

33. More Practice Scores on the IQ-Test have an approximately normal distribution. The mean score is 100 and the standard deviation is 15. What percent of the population has a score greater than 130? The problem is asking for the percent greater than 2σ to the right of the mean. Since +2σ of the mean is 95%, then 5% remains in the two tails. Only the right tail is of interest. We can divide the 5% in half because the normal curve is symmetric. 2.5% of the population scores higher than 115 55 70 85 100 115 130 145 IQ Score -3σ -2σ -1σ 0 1σ 2σ 3σ 95%

34. Practice Again A data set that follows a bell-shape and symmetrical distribution has a mean equal to 75 and a standard deviation equal to 10. 1.What range of values centered around the mean would represent 95% of the values? 2.What percent of the values are above 55? 3.What percent falls between 85 and 65? 4.What percent falls between 55 and 65?

35. Answers 45 55 65 75 85 95 105 1.The range of values would be from 55 to 95, since 95% of the data fall within 2 standard deviations. 2.Above 75 is 50%. The two standard deviations to left are 34% and 13.5%. The total percent above 75 is 50+34+13.5 = 97.5% 3.Both 85 and 65 are 1 standard deviation from the mean. Therefore 68% of the data falls between 65 and 85. 4.Between 65 and 85 is 68% of the data. Between 55 and 65 is 13.5%. Total % = 68+13.5=81.5% of the data.

36. Slide 6- 36 Are You Normal? How Can You Tell? When you actually have your own data, you must check to see whether a Normal model is reasonable. Looking at a histogram of the data is a good way to check that the underlying distribution is roughly unimodal and symmetric.

37. Slide 6- 37 Before using the normal model you must check to see if your data is unimodal and symmetric. A quick look at your data in histogram will allow you to verify this. If you use a normal model to when your data is not approximately normal, then your probabilities will not be accurate.

38. Revisit your diagnostic problem set that you completed prior to starting this module or take it for the first time. Assess your knowledge and understanding of the targeted outcome skills. Go to SolutionsSolutions Go to final slide for a quick survey.

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