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Applied Math 40S March 19, 2008

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The Normal Distribution and it's properties.

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Applied Math 40S March 19, 2008

1. 1. The Normal Curve Curve by JasonUnbound
2. 2. 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?
3. 3. 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? (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?
4. 4. HOMEWORK Four hundred people were surveyed No. of Videos Returned No. of Persons to find how many videos they had 1 28 rented during the last month. 2 102 Determine the mean and median of the 3 160 frequency distribution shown below, 4 70 and draw a probability distribution 5 25 histogram. Also, determine the mode 6 13 by inspecting the frequency 7 0 distribution and the histogram. 8 2
5. 5. HOMEWORK The table shows the weights weight interval mean interval # of infants (in pounds) of 125 newborn 3.5 to 4.5 4 4 infants. The first column 4.5 to 5.5 5 11 shows the weight interval, 5.5 to 6.5 6 19 the second column the 6.5 to 7.5 7 33 average weight within each 7.5 to 8.5 8 29 weight interval, and the third 8.5 to 9.5 9 17 column the number of 9.5 to 10.5 10 8 newborn infants at each 10.5 to 11.5 11 4 weight. Total 125 (a) Calculate the mean weight and standard deviation. (b) Calculate the weight of an infant at one standard deviation below the mean weight, and one standard deviation above the mean. (c) Determine the number of infants whose weights are within one standard deviation of the mean weight. (d) What percent of the infants have weights that are within one standard deviation of the mean weight?
6. 6. 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?
7. 7. HOMEWORK The contents in the cans of several cases of soft drinks were tested. The mean contents per can is 356 mL, and the standard deviation is 1.5 mL. (a) Two cans were randomly selected and tested. One can held 358 mL, and the other can 352 mL. Calculate the z-score of each. (b) Two other cans had z-scores of -3 and 1.85. How many mL did each contain?
8. 8. HOMEWORK 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? (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?
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: • the heights and weights of adult males in North America • the times for athletes to run 5000 metres • the speed of cars on a busy highway • the weights of loonies produced at the Winnipeg Mint Note that all the examples represent continuous data.
11. 11. 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. • The area under the curve always equals one. • The x-axis is an asymptote for the curve. Frequency Scores Interactivate Normal Distribution
12. 12. 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 standard deviations of the mean.
13. 13. 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.
14. 14. Properties of a Normal Distribution The shape of any normal distribution curve is determined by: • the mean (μ) • the standard deviation (σ) Changing the mean will shift the graph horizontally. Changing the standard deviation will change the shape of the curve, making it narrow or wide.
15. 15. Properties of a Normal Distribution The data are continuous and distributed evenly around the mean, and the graph created by the data is a bell-shaped curve, as shown in the examples below. These curves represent data sets that have the same mean, but different standard deviations. Which one has a larger standard deviation (σ)? How can you tell?
16. 16. 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
17. 17. 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.
18. 18. The chart shows the sizes of pants sold in one week at Dan's Clothing Shop. 38 34 42 40 42 32 30 34 40 38 40 38 36 42 44 42 38 36 36 42 36 46 40 38 40 36 44 36 38 34 38 40 Determine whether the data approximate the normal distribution.
19. 19. 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.
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 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.
22. 22. 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