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# Applied 40S March 27, 2009

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The standard normal curve. Properties of normal distributions.

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### Applied 40S March 27, 2009

1. 1. The Normal Curve Curve by JasonUnbound
2. 2. z-scores A z-score may also be described by the following formula: z = the z-score (standardized score) x = a number in the distribution μ = the population mean σ = standard deviation for the population
3. 3. Standardizing Two Sets of Scores Two consumer groups, Vancouver Halifax one in Vancouver and Cereal Brand Rating Cereal Brand Rating one in Halifax, recently A 1 P 25 tested ﬁve brands of B 10 Q 35 breakfast cereal for C 15 R 45 taste appeal. Each D 21 S 50 consumer group used E 28 T 70 a different rating system. HOMEWORK Use z-scores to determine which cereal has the higher taste appeal rating. T has the higher taste appeal rating.
4. 4. The table shows the lengths in millimetres of 52 arrowheads. 16 16 17 17 18 18 18 18 19 20 20 21 21 21 22 22 22 23 23 23 24 24 25 25 25 26 26 26 26 27 27 27 27 27 28 28 28 28 29 30 30 30 30 30 30 31 33 33 34 35 39 40 HOMEWORK (a) Calculate the mean length and the standard deviation. (b) Determine the lengths of arrowheads one standard deviation below and one standard deviation above the mean. (c) How many arrowheads are within one standard deviation of the mean? (d) What percent of the arrowheads are within one standard deviation of the mean length?
5. 5. HOMEWORK The table shows the lengths in millimetres of 52 arrowheads. 16 16 17 17 18 18 18 18 19 20 20 21 21 21 22 22 22 23 23 23 24 24 25 25 25 26 26 26 26 27 27 27 27 27 28 28 28 28 29 30 30 30 30 30 30 31 33 33 34 35 39 40 (a) Calculate the mean length (b) Determine the lengths of arrowheads and the standard deviation. one standard deviation below and one standard deviation above the mean. (c) How many arrowheads (d) What percent of the arrowheads are within one standard are within one standard deviation deviation of the mean? of the mean length?
6. 6. Using Z-Score, Mean, and Standard Deviation to Calculate the Real Score Numerous packages of raisins were weighed. The mean mass was 1600 grams, and the standard deviation was 40 grams. Trudy bought a package that had a z-score of -1.6. What was the mass of Trudy's package of raisins? HOMEWORK
7. 7. HOMEWORK A survey was conducted at DMCI to determine the number of music CDs each student owned. The results of the survey showed that the average number of CDs per student was 73 with a standard deviation of 24. After the scores were standardized, the people doing the survey discovered that DJ Chunky had a z-score rating of 2.9. How many CDs does Chunky have?
8. 8. North American women have a mean height of 161.5 cm and a standard deviation of 6.3 cm. (a) A car designer designs car seats to ﬁt women taller than 159.0 cm. What is the z-score of a woman who is 159.0 cm tall? z = -0.3968 (b) The manufacturer designs the seats to ﬁt women with a maximum z-score of 2.8. How tall is a woman with a z-score of 2.8? 179.14 = x
9. 9. The Normal Distribution A Normal Distribution is a frequency distribution that can be represented by a symmetrical bell-shaped curve which shows that most of the data are concentrated around the centre (i.e., mean) of the distribution. The mean, median, and mode are all equal. Since the median is the same as the mean, 50 percent of the data are lower than the mean, and 50 percent are higher. The frequency distribution showing light bulb life, for example, shows that the mean is 970 hours, and the hours of life for all the bulbs are spread uniformly about the mean.
10. 10. The Normal Distribution The diagram above represents a normal distribution. In real life, the data would never ﬁt a normal distribution perfectly. There are, however, many situations where data do approximate a normal distribution. Some examples would include: (Note that all the examples represent continuous data.)
11. 11. • the heights and weights of adult males in North America World Strong Man Competition 2007 by ﬂickr user highstrungloner
12. 12. • the times for athletes to swim 5000 metres United States Olympic Triathlon Trials by ﬂickr user Diamondduste
13. 13. • the speed of cars on a busy highway by ﬂickr user El Fotopakismo
14. 14. • the weights of quarters produced at the Winnipeg Mint IMG_3677.JPG by ﬂickr user JonBen
15. 15. The diagram shows a normal distribution with a mean of 28 and a standard deviation of 4. The values represent the number of standard deviations above and below the mean. Replace the numbers with raw scores.
16. 16. Properties of a Normal Distribution The 68-95-99 Rule Generally speaking, approximately: • 68% of all the data in a normal distribution lie within the 1 standard deviation of the mean, • 95% of all the data lie within 2 standard deviations of the mean, and • 99.7% of all the data lie within 3 standard deviations of the mean.
17. 17. Properties of a Normal Distribution The curve is symmetrical about the mean. Most of the data are relatively close to the mean, and the number of data decrease as you get farther from the mean.
18. 18. More Properties of a Normal Distribution • 99.7% of all the data lies within approximately 3 standard deviations of the mean. • All normal distributions are symetrical about the mean. • Each value of mean and standard deviation determines a different normal distributions. (see below) • The area under the curve always equals one. • The x-axis is an asymptote for the curve. Frequency Scores Interactivate Normal Distribution
19. 19. The data below shows the ages in years of 30 trees in an area of natural vegetation. 37 15 34 26 25 38 19 22 21 28 42 18 27 32 19 17 29 28 24 35 35 20 23 36 21 39 16 40 18 41 Determine whether the data approximate the normal distribution. USING the 68 -95-97 RULE
20. 20. HOMEWORK The following are the number of steak dinners served on 50 consecutive Sundays at a restaurant. 41 52 46 42 46 36 44 68 58 44 49 48 48 65 52 50 45 72 45 43 47 49 57 44 48 49 45 47 48 43 45 56 61 54 51 47 42 53 44 45 58 55 43 63 38 42 43 46 49 47 Draw a suitable histogram that has ﬁve bars.
21. 21. HOMEWORK The frequency table shows the ages of all the students in Senior 4 Math at Newberry High. Find the mean, μ. Then calculate the percent of students older than the mean age. How does this compare to the percent of students older than the mean age if the distribution were a normal distribution? Based on this answer, does it seem that the students' ages approximate a normal distribution? Age of Student 15 16 17 18 19 20 21 22 # of Students 1 7 42 24 7 4 2 1
22. 22. Now let's try a problem involving Grouped Data A machine is used to ﬁll bags with beans. The machine is set to add 10 kilograms of beans to each bag. The table shows the weights of 277 bags that were randomly selected. wt in kg 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5 # of bags 1 3 13 25 41 66 52 41 25 7 3 (a) Are the weights normally distributed? How do you know? (b) Do you think that using the machine is acceptable and fair to the customers? Explain your reasoning.